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Properties

Label 1-2805-2805.1238-r0-0-0
Degree $1$
Conductor $2805$
Sign $-0.342 + 0.939i$
Analytic cond. $13.0263$
Root an. cond. $13.0263$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(1\)
Conductor: \(2805\)    =    \(3 \cdot 5 \cdot 11 \cdot 17\)
Sign: $-0.342 + 0.939i$
Analytic conductor: \(13.0263\)
Root analytic conductor: \(13.0263\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2805} (1238, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2805,\ (0:\ ),\ -0.342 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.747612613 + 2.497592963i\)
\(L(\frac12)\) \(\approx\) \(1.747612613 + 2.497592963i\)
\(L(1)\) \(\approx\) \(1.755691987 + 0.7403848482i\)
\(L(1)\) \(\approx\) \(1.755691987 + 0.7403848482i\)

Euler product

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

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 $Z$-function along the critical line