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Properties

Label 1-4235-4235.1537-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.480 + 0.876i$
Analytic cond. $19.6672$
Root an. cond. $19.6672$
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  + (−0.299 − 0.953i)2-s + (0.406 − 0.913i)3-s + (−0.820 + 0.572i)4-s + (−0.993 − 0.113i)6-s + (0.791 + 0.610i)8-s + (−0.669 − 0.743i)9-s + (0.189 + 0.981i)12-s + (−0.389 − 0.921i)13-s + (0.345 − 0.938i)16-s + (−0.263 − 0.964i)17-s + (−0.508 + 0.861i)18-s + (−0.625 − 0.780i)19-s + (−0.618 + 0.786i)23-s + (0.879 − 0.475i)24-s + (−0.761 + 0.647i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (−0.299 − 0.953i)2-s + (0.406 − 0.913i)3-s + (−0.820 + 0.572i)4-s + (−0.993 − 0.113i)6-s + (0.791 + 0.610i)8-s + (−0.669 − 0.743i)9-s + (0.189 + 0.981i)12-s + (−0.389 − 0.921i)13-s + (0.345 − 0.938i)16-s + (−0.263 − 0.964i)17-s + (−0.508 + 0.861i)18-s + (−0.625 − 0.780i)19-s + (−0.618 + 0.786i)23-s + (0.879 − 0.475i)24-s + (−0.761 + 0.647i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.480 + 0.876i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.480 + 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4018779795 - 0.2379535913i\)
\(L(\frac12)\) \(\approx\) \(-0.4018779795 - 0.2379535913i\)
\(L(1)\) \(\approx\) \(0.4536313723 - 0.6137961085i\)
\(L(1)\) \(\approx\) \(0.4536313723 - 0.6137961085i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.299 - 0.953i)T \)
3 \( 1 + (0.406 - 0.913i)T \)
13 \( 1 + (-0.389 - 0.921i)T \)
17 \( 1 + (-0.263 - 0.964i)T \)
19 \( 1 + (-0.625 - 0.780i)T \)
23 \( 1 + (-0.618 + 0.786i)T \)
29 \( 1 + (-0.998 - 0.0570i)T \)
31 \( 1 + (0.483 - 0.875i)T \)
37 \( 1 + (-0.956 - 0.290i)T \)
41 \( 1 + (-0.198 - 0.980i)T \)
43 \( 1 + (-0.281 - 0.959i)T \)
47 \( 1 + (0.508 + 0.861i)T \)
53 \( 1 + (0.938 - 0.345i)T \)
59 \( 1 + (0.948 - 0.318i)T \)
61 \( 1 + (0.217 + 0.976i)T \)
67 \( 1 + (-0.0950 - 0.995i)T \)
71 \( 1 + (0.774 + 0.633i)T \)
73 \( 1 + (0.244 - 0.969i)T \)
79 \( 1 + (-0.432 - 0.901i)T \)
83 \( 1 + (0.441 + 0.897i)T \)
89 \( 1 + (0.888 - 0.458i)T \)
97 \( 1 + (-0.999 - 0.0285i)T \)
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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

−18.944363891595646216103818654388, −18.20019505396460892704697019086, −17.25689178824452637466589035549, −16.77032545126756707538461451628, −16.30676589263214677909828250325, −15.563053135077306127801871479990, −14.842290179254975845621360146409, −14.50824729470878060398037322272, −13.82464120265170577507767596299, −13.05461904150437031022552695190, −12.173567677042888137150728820612, −11.16846847302676586689593263022, −10.355073987468934309543815799209, −9.97075567800708896453032209998, −9.168811962958649652707560793158, −8.442221549102883495180110480962, −8.150608766430806902529832093303, −7.0594467504323158353710223737, −6.38859270429186443022214561669, −5.61474813405033636328088827210, −4.83046219957649993454506897627, −4.14367044233780201309749639272, −3.6364985677426075485134346294, −2.3288290195994529502176450063, −1.52867837466813281924994119216, 0.15767712107706178789516916541, 0.88318255155170056501868156616, 2.08222864753687019367296688429, 2.38587217599645130122980849669, 3.32135626479926277962687426899, 3.974604286680008297162855551302, 5.08995139431477457546097360989, 5.7379428243555197909357632042, 6.93023074640489029735755470570, 7.49167542189910304879445975416, 8.13849975398984498027849402572, 8.92936077394082187744405325431, 9.42727940514772151809502624170, 10.28478082243198759953297268468, 11.04973867449034797739867285480, 11.80422643775607732814879442205, 12.24067535295798412827281046334, 13.11636846908394100994800958662, 13.45753159158570565867072264340, 14.11954453219263694993730175115, 15.00091268300988598667577198609, 15.6840642900400153189487167936, 16.838856895357820828680006821079, 17.50344015301681041805807052957, 17.84693688807089431090751074924

Graph of the $Z$-function along the critical line