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Properties

Label 1-55e2-3025.796-r0-0-0
Degree $1$
Conductor $3025$
Sign $-0.932 - 0.362i$
Analytic cond. $14.0480$
Root an. cond. $14.0480$
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  + (−0.564 + 0.825i)2-s + (0.309 − 0.951i)3-s + (−0.362 − 0.931i)4-s + (0.610 + 0.791i)6-s + (0.198 + 0.980i)7-s + (0.974 + 0.226i)8-s + (−0.809 − 0.587i)9-s + (−0.998 + 0.0570i)12-s + (−0.362 − 0.931i)13-s + (−0.921 − 0.389i)14-s + (−0.736 + 0.676i)16-s + (−0.466 + 0.884i)17-s + (0.941 − 0.336i)18-s + (−0.466 − 0.884i)19-s + (0.993 + 0.113i)21-s + ⋯
L(s)  = 1  + (−0.564 + 0.825i)2-s + (0.309 − 0.951i)3-s + (−0.362 − 0.931i)4-s + (0.610 + 0.791i)6-s + (0.198 + 0.980i)7-s + (0.974 + 0.226i)8-s + (−0.809 − 0.587i)9-s + (−0.998 + 0.0570i)12-s + (−0.362 − 0.931i)13-s + (−0.921 − 0.389i)14-s + (−0.736 + 0.676i)16-s + (−0.466 + 0.884i)17-s + (0.941 − 0.336i)18-s + (−0.466 − 0.884i)19-s + (0.993 + 0.113i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.932 - 0.362i$
Analytic conductor: \(14.0480\)
Root analytic conductor: \(14.0480\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (796, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3025,\ (0:\ ),\ -0.932 - 0.362i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06036982317 - 0.3219155534i\)
\(L(\frac12)\) \(\approx\) \(0.06036982317 - 0.3219155534i\)
\(L(1)\) \(\approx\) \(0.7040106824 + 0.02924411189i\)
\(L(1)\) \(\approx\) \(0.7040106824 + 0.02924411189i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.564 + 0.825i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.198 + 0.980i)T \)
13 \( 1 + (-0.362 - 0.931i)T \)
17 \( 1 + (-0.466 + 0.884i)T \)
19 \( 1 + (-0.466 - 0.884i)T \)
23 \( 1 + (0.198 - 0.980i)T \)
29 \( 1 + (0.696 - 0.717i)T \)
31 \( 1 + (0.841 - 0.540i)T \)
37 \( 1 + (-0.254 - 0.967i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (-0.0285 + 0.999i)T \)
53 \( 1 + (0.198 + 0.980i)T \)
59 \( 1 + (-0.0285 + 0.999i)T \)
61 \( 1 + (0.610 - 0.791i)T \)
67 \( 1 + (-0.564 + 0.825i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (0.415 + 0.909i)T \)
79 \( 1 + (0.0855 - 0.996i)T \)
83 \( 1 + (-0.870 + 0.491i)T \)
89 \( 1 + (-0.466 + 0.884i)T \)
97 \( 1 + (0.974 + 0.226i)T \)
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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

−19.705008455429140593102613048611, −18.80971147894167475225526248074, −17.95065671906834152136155619378, −17.195595255710146783471893488425, −16.64180589613424134765862122943, −16.16469768828560167367431843809, −15.211496262564180804885524098324, −14.23532054002410417794108182310, −13.79649805885010791008678044454, −13.12208317317734848525179780151, −11.87534475893656299270180358896, −11.53981746277489918206427138739, −10.68727805041624259385811960367, −10.046579772943253689824685807067, −9.645778561014893545355552582090, −8.71683953801277407953223844435, −8.19429975856489491187449793354, −7.27864355062226898693010574834, −6.545038661345227801008378314357, −4.87207251133363540818821260730, −4.73152779968745941181153849432, −3.666524931007069382649164395840, −3.21803183044785892274638984737, −2.13554218254859908608330679811, −1.318719376865836378003843503640, 0.121423084297680030131784835614, 1.17691958193807783807418437022, 2.255336043210664429511176754701, 2.70987728871438434168896913893, 4.18887321513236745831640983459, 5.15001042953062242598215759735, 5.90476097721614497350608423708, 6.47519708327060828525786747249, 7.201771325729046953643130600064, 8.11399298733383512190787188539, 8.504405574542548237448277105068, 9.065418217183053400988837530197, 10.0633685098150449508492950255, 10.84183934155340987717471565556, 11.7316516185935357652717329255, 12.580238680225495516189602892966, 13.13580438040998956889065695645, 13.94033452622944394676954222186, 14.74625026478654641314763545906, 15.24722301263520535283240388475, 15.724801598276627407213438488895, 16.9598359747240196506165404757, 17.48570864452966824605727482588, 17.906975653743019231787768338316, 18.77771197647297000405407890146

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