Properties

Label 1-2805-2805.1238-r0-0-0
Degree 11
Conductor 28052805
Sign 0.342+0.939i-0.342 + 0.939i
Analytic cond. 13.026313.0263
Root an. cond. 13.026313.0263
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.987 + 0.156i)2-s + (0.951 + 0.309i)4-s + (0.0784 + 0.996i)7-s + (0.891 + 0.453i)8-s + (−0.809 + 0.587i)13-s + (−0.0784 + 0.996i)14-s + (0.809 + 0.587i)16-s + (0.891 + 0.453i)19-s + (0.382 + 0.923i)23-s + (−0.891 + 0.453i)26-s + (−0.233 + 0.972i)28-s + (−0.760 − 0.649i)29-s + (−0.972 + 0.233i)31-s + (0.707 + 0.707i)32-s + (0.760 + 0.649i)37-s + (0.809 + 0.587i)38-s + ⋯
L(s)  = 1  + (0.987 + 0.156i)2-s + (0.951 + 0.309i)4-s + (0.0784 + 0.996i)7-s + (0.891 + 0.453i)8-s + (−0.809 + 0.587i)13-s + (−0.0784 + 0.996i)14-s + (0.809 + 0.587i)16-s + (0.891 + 0.453i)19-s + (0.382 + 0.923i)23-s + (−0.891 + 0.453i)26-s + (−0.233 + 0.972i)28-s + (−0.760 − 0.649i)29-s + (−0.972 + 0.233i)31-s + (0.707 + 0.707i)32-s + (0.760 + 0.649i)37-s + (0.809 + 0.587i)38-s + ⋯

Functional equation

Λ(s)=(2805s/2ΓR(s)L(s)=((0.342+0.939i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2805s/2ΓR(s)L(s)=((0.342+0.939i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 2805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 28052805    =    3511173 \cdot 5 \cdot 11 \cdot 17
Sign: 0.342+0.939i-0.342 + 0.939i
Analytic conductor: 13.026313.0263
Root analytic conductor: 13.026313.0263
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2805(1238,)\chi_{2805} (1238, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2805, (0: ), 0.342+0.939i)(1,\ 2805,\ (0:\ ),\ -0.342 + 0.939i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.747612613+2.497592963i1.747612613 + 2.497592963i
L(12)L(\frac12) \approx 1.747612613+2.497592963i1.747612613 + 2.497592963i
L(1)L(1) \approx 1.755691987+0.7403848482i1.755691987 + 0.7403848482i
L(1)L(1) \approx 1.755691987+0.7403848482i1.755691987 + 0.7403848482i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1 1
5 1 1
11 1 1
17 1 1
good2 1+(0.987+0.156i)T 1 + (0.987 + 0.156i)T
7 1+(0.0784+0.996i)T 1 + (0.0784 + 0.996i)T
13 1+(0.809+0.587i)T 1 + (-0.809 + 0.587i)T
19 1+(0.891+0.453i)T 1 + (0.891 + 0.453i)T
23 1+(0.382+0.923i)T 1 + (0.382 + 0.923i)T
29 1+(0.7600.649i)T 1 + (-0.760 - 0.649i)T
31 1+(0.972+0.233i)T 1 + (-0.972 + 0.233i)T
37 1+(0.760+0.649i)T 1 + (0.760 + 0.649i)T
41 1+(0.760+0.649i)T 1 + (-0.760 + 0.649i)T
43 1+(0.7070.707i)T 1 + (0.707 - 0.707i)T
47 1+(0.3090.951i)T 1 + (-0.309 - 0.951i)T
53 1+(0.987+0.156i)T 1 + (0.987 + 0.156i)T
59 1+(0.891+0.453i)T 1 + (-0.891 + 0.453i)T
61 1+(0.2330.972i)T 1 + (0.233 - 0.972i)T
67 1+iT 1 + iT
71 1+(0.5220.852i)T 1 + (-0.522 - 0.852i)T
73 1+(0.649+0.760i)T 1 + (-0.649 + 0.760i)T
79 1+(0.522+0.852i)T 1 + (-0.522 + 0.852i)T
83 1+(0.156+0.987i)T 1 + (0.156 + 0.987i)T
89 1iT 1 - iT
97 1+(0.972+0.233i)T 1 + (-0.972 + 0.233i)T
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   L(s)=p (1αpps)1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−19.29005599143245831947567237066, −18.30133559192630329578693312630, −17.484768374200144597071364888266, −16.646947607608293604358645428660, −16.23311343403678241767293038389, −15.26523134642727058941636989960, −14.57494597277743497573076073062, −14.14661730825170860880124879559, −13.16776755992618267883984997764, −12.843524376000316456899829762407, −11.962121524536453986848493746987, −11.13767490949661541601410357490, −10.609499354769643921824210318393, −9.89157451398277797961719767069, −9.00453064284233793578470443288, −7.54731032106464567337278187417, −7.48278568215130823889557013092, −6.52377342541154874245364012216, −5.59729690956036958108611152695, −4.89921992363889334090914673119, −4.21339848827916130625363895804, −3.352642730192328615779247912274, −2.66088006958059531405158220902, −1.62372994748377228493667104060, −0.61213802723578970963641542671, 1.49651403491684209007917603423, 2.23290540138838999589932592607, 3.05188622076704374659593479936, 3.84022286781422509748826718155, 4.8071599335830228319115550056, 5.45523535174504971398632522914, 6.004305195498494519531107125990, 7.04350249434553022440353541593, 7.56218653761994928146881660496, 8.49810447618871667598682942941, 9.4071961027918653221509279619, 10.091102310576613630048375674728, 11.30944178297334752809808576639, 11.65892875680220954599999323732, 12.32882359642262614020072778658, 13.057491488632453499084006176829, 13.79355093387614589519684593868, 14.54443377652444460663180218587, 15.127467396128349192614782837000, 15.67158655005980929004627275905, 16.56421325346518207030483818876, 17.02877659297445331126598761800, 18.07075508038130769714087016678, 18.76898577293055198256677292180, 19.54484874952542853229609617889

Graph of the ZZ-function along the critical line