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Properties

Label 1-3215-3215.37-r0-0-0
Degree $1$
Conductor $3215$
Sign $-0.950 + 0.311i$
Analytic cond. $14.9303$
Root an. cond. $14.9303$
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  + (−0.261 + 0.965i)2-s + (0.771 + 0.636i)3-s + (−0.863 − 0.504i)4-s + (−0.815 + 0.578i)6-s + (−0.777 − 0.628i)7-s + (0.712 − 0.701i)8-s + (0.189 + 0.981i)9-s + (0.827 + 0.562i)11-s + (−0.345 − 0.938i)12-s + (0.918 − 0.395i)13-s + (0.810 − 0.586i)14-s + (0.491 + 0.870i)16-s + (−0.609 + 0.792i)17-s + (−0.997 − 0.0733i)18-s + (−0.227 − 0.973i)19-s + ⋯
L(s)  = 1  + (−0.261 + 0.965i)2-s + (0.771 + 0.636i)3-s + (−0.863 − 0.504i)4-s + (−0.815 + 0.578i)6-s + (−0.777 − 0.628i)7-s + (0.712 − 0.701i)8-s + (0.189 + 0.981i)9-s + (0.827 + 0.562i)11-s + (−0.345 − 0.938i)12-s + (0.918 − 0.395i)13-s + (0.810 − 0.586i)14-s + (0.491 + 0.870i)16-s + (−0.609 + 0.792i)17-s + (−0.997 − 0.0733i)18-s + (−0.227 − 0.973i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3215 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3215 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3215\)    =    \(5 \cdot 643\)
Sign: $-0.950 + 0.311i$
Analytic conductor: \(14.9303\)
Root analytic conductor: \(14.9303\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3215} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3215,\ (0:\ ),\ -0.950 + 0.311i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2300529351 + 1.439925193i\)
\(L(\frac12)\) \(\approx\) \(0.2300529351 + 1.439925193i\)
\(L(1)\) \(\approx\) \(0.7943354263 + 0.6890264820i\)
\(L(1)\) \(\approx\) \(0.7943354263 + 0.6890264820i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
643 \( 1 \)
good2 \( 1 + (-0.261 + 0.965i)T \)
3 \( 1 + (0.771 + 0.636i)T \)
7 \( 1 + (-0.777 - 0.628i)T \)
11 \( 1 + (0.827 + 0.562i)T \)
13 \( 1 + (0.918 - 0.395i)T \)
17 \( 1 + (-0.609 + 0.792i)T \)
19 \( 1 + (-0.227 - 0.973i)T \)
23 \( 1 + (-0.354 + 0.935i)T \)
29 \( 1 + (-0.900 + 0.435i)T \)
31 \( 1 + (-0.837 + 0.545i)T \)
37 \( 1 + (0.958 + 0.284i)T \)
41 \( 1 + (0.994 + 0.107i)T \)
43 \( 1 + (0.669 - 0.742i)T \)
47 \( 1 + (0.155 - 0.987i)T \)
53 \( 1 + (0.512 + 0.858i)T \)
59 \( 1 + (0.349 - 0.936i)T \)
61 \( 1 + (-0.574 + 0.818i)T \)
67 \( 1 + (-0.175 - 0.984i)T \)
71 \( 1 + (0.658 - 0.752i)T \)
73 \( 1 + (-0.906 + 0.421i)T \)
79 \( 1 + (0.980 - 0.194i)T \)
83 \( 1 + (-0.545 + 0.837i)T \)
89 \( 1 + (-0.256 + 0.966i)T \)
97 \( 1 + (-0.335 - 0.941i)T \)
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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

−18.586675296507537062813335251795, −18.44346716350893892791555949517, −17.52344940269276244020748818053, −16.46433501403964194521035410721, −16.07721037475048019178099775836, −14.78921293500728356964741605815, −14.302452335289076134731930859440, −13.51540214205776975031695954767, −12.94808057914900418716991533759, −12.386549819905143800581007148663, −11.578954181896977232409168700630, −11.0700211322683138180326199905, −9.90430737752598931963707327098, −9.23177552782516208388264366432, −8.90344157997075471146522172213, −8.14806238572089039328759417335, −7.31517173916105071128736339876, −6.27943520405499319334922048543, −5.815626734838908804911843404414, −4.19316036983482033007192336287, −3.84033703663499522545398112866, −2.90997044638465358067629551116, −2.29431501909696160204042785636, −1.47219174620960540010559299777, −0.48342719951596132233907768678, 1.06989789105676767989409067391, 2.0932070698693158705783703686, 3.47994003140015094117927735835, 3.89137012212413534003842609721, 4.58131260724006657299391366222, 5.61522006786181723050423113203, 6.374835905018766618519580897532, 7.18114551067389826284864927633, 7.71418678039823903072754949219, 8.76940357323879095005970829773, 9.12438284063600500527887999488, 9.773348658784376550174801006032, 10.59444270581403619423264675329, 11.10638933884321487607661795590, 12.65537445457458652253657055360, 13.229252702886653525786699784791, 13.77183309299309054817097822686, 14.50293082211283494289597134217, 15.33910065130230606539043056186, 15.53870808867892964677958858657, 16.47919480122315391667706889063, 16.91678817813185205630266811310, 17.72423040213029040549408575961, 18.43967904423702192959222041612, 19.49529244640594738876533657140

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