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Properties

Label 1-2608-2608.1029-r0-0-0
Degree $1$
Conductor $2608$
Sign $-0.912 + 0.409i$
Analytic cond. $12.1115$
Root an. cond. $12.1115$
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  + (−0.0387 + 0.999i)3-s + (0.727 − 0.686i)5-s + (−0.323 − 0.946i)7-s + (−0.996 − 0.0774i)9-s + (0.845 + 0.533i)11-s + (−0.998 + 0.0581i)13-s + (0.657 + 0.753i)15-s + (−0.835 − 0.549i)17-s + (−0.516 − 0.856i)19-s + (0.957 − 0.286i)21-s + (−0.173 + 0.984i)23-s + (0.0581 − 0.998i)25-s + (0.116 − 0.993i)27-s + (0.778 + 0.627i)29-s + (−0.286 + 0.957i)31-s + ⋯
L(s)  = 1  + (−0.0387 + 0.999i)3-s + (0.727 − 0.686i)5-s + (−0.323 − 0.946i)7-s + (−0.996 − 0.0774i)9-s + (0.845 + 0.533i)11-s + (−0.998 + 0.0581i)13-s + (0.657 + 0.753i)15-s + (−0.835 − 0.549i)17-s + (−0.516 − 0.856i)19-s + (0.957 − 0.286i)21-s + (−0.173 + 0.984i)23-s + (0.0581 − 0.998i)25-s + (0.116 − 0.993i)27-s + (0.778 + 0.627i)29-s + (−0.286 + 0.957i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2608 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2608\)    =    \(2^{4} \cdot 163\)
Sign: $-0.912 + 0.409i$
Analytic conductor: \(12.1115\)
Root analytic conductor: \(12.1115\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2608} (1029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2608,\ (0:\ ),\ -0.912 + 0.409i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09861557726 + 0.4610349545i\)
\(L(\frac12)\) \(\approx\) \(0.09861557726 + 0.4610349545i\)
\(L(1)\) \(\approx\) \(0.8676023594 + 0.1531816540i\)
\(L(1)\) \(\approx\) \(0.8676023594 + 0.1531816540i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
163 \( 1 \)
good3 \( 1 + (-0.0387 + 0.999i)T \)
5 \( 1 + (0.727 - 0.686i)T \)
7 \( 1 + (-0.323 - 0.946i)T \)
11 \( 1 + (0.845 + 0.533i)T \)
13 \( 1 + (-0.998 + 0.0581i)T \)
17 \( 1 + (-0.835 - 0.549i)T \)
19 \( 1 + (-0.516 - 0.856i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.778 + 0.627i)T \)
31 \( 1 + (-0.286 + 0.957i)T \)
37 \( 1 + (-0.918 + 0.396i)T \)
41 \( 1 + (-0.952 - 0.305i)T \)
43 \( 1 + (-0.378 + 0.925i)T \)
47 \( 1 + (-0.360 + 0.932i)T \)
53 \( 1 + (-0.342 - 0.939i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (0.448 - 0.893i)T \)
67 \( 1 + (0.672 + 0.740i)T \)
71 \( 1 + (0.135 + 0.990i)T \)
73 \( 1 + (0.740 + 0.672i)T \)
79 \( 1 + (-0.565 - 0.824i)T \)
83 \( 1 + (-0.192 + 0.981i)T \)
89 \( 1 + (-0.533 - 0.845i)T \)
97 \( 1 + (-0.431 + 0.902i)T \)
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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

−19.00918228554746809660325365600, −18.45104833121480178778578468752, −17.71844826244176782224332118925, −17.067994068771503229599179314093, −16.52279167693498671858529795852, −15.11508565231895373440446894771, −14.842606999562128162352239460480, −13.96276654909160416226638992146, −13.431233655645976905338345028613, −12.470301353166974120030046990011, −12.09558972808780182105367939635, −11.25653261369418516252467429816, −10.39962089704613295527965908832, −9.60683715964822930868544604559, −8.715896053516505232687287629174, −8.23746279072627285426930598834, −7.0369051916668300090411296282, −6.51033669810730123948781491505, −5.98443630503794805914878033563, −5.26163051816100519350638528835, −3.93889009349324697919111356975, −2.90272589653318728325284695236, −2.22296197020977815660697418976, −1.70113352474559341389690827565, −0.14128561335192341145466652153, 1.19727586168772621935104402810, 2.245300161759424999300749674848, 3.24327426900328525550128314395, 4.154146637404285008324476480555, 4.84751297912644713375472998256, 5.24132641676287912812983482567, 6.65131251574180888397435293208, 6.86655659815490017251084739850, 8.25200107377163646172006247189, 9.00176438084194407532028114999, 9.70194227101106708291232539584, 9.98718534305383868362544851857, 10.94297210433555827265435634019, 11.68141798688631436853189302507, 12.56374836616151755107931872100, 13.31731843156582276421735432758, 14.136074782363741590615800769383, 14.53010682974270483035324737257, 15.631640301111765881247241997185, 16.10576434088624966936759217293, 16.9530526688893162857398450648, 17.48646051537708763951405957419, 17.68788767893092533605905525047, 19.35989349443099237408185188360, 19.9017209341451290021796370177

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