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Properties

Label 1-4235-4235.3092-r0-0-0
Degree $1$
Conductor $4235$
Sign $0.957 + 0.288i$
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  + (0.458 − 0.888i)2-s + (−0.866 − 0.5i)3-s + (−0.580 − 0.814i)4-s + (−0.841 + 0.540i)6-s + (−0.989 + 0.142i)8-s + (0.5 + 0.866i)9-s + (0.0950 + 0.995i)12-s + (0.909 + 0.415i)13-s + (−0.327 + 0.945i)16-s + (−0.690 − 0.723i)17-s + (0.998 − 0.0475i)18-s + (0.723 + 0.690i)19-s + (0.945 + 0.327i)23-s + (0.928 + 0.371i)24-s + (0.786 − 0.618i)26-s − i·27-s + ⋯
L(s)  = 1  + (0.458 − 0.888i)2-s + (−0.866 − 0.5i)3-s + (−0.580 − 0.814i)4-s + (−0.841 + 0.540i)6-s + (−0.989 + 0.142i)8-s + (0.5 + 0.866i)9-s + (0.0950 + 0.995i)12-s + (0.909 + 0.415i)13-s + (−0.327 + 0.945i)16-s + (−0.690 − 0.723i)17-s + (0.998 − 0.0475i)18-s + (0.723 + 0.690i)19-s + (0.945 + 0.327i)23-s + (0.928 + 0.371i)24-s + (0.786 − 0.618i)26-s − i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $0.957 + 0.288i$
Analytic conductor: \(19.6672\)
Root analytic conductor: \(19.6672\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (3092, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4235,\ (0:\ ),\ 0.957 + 0.288i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8738059814 + 0.1287929416i\)
\(L(\frac12)\) \(\approx\) \(0.8738059814 + 0.1287929416i\)
\(L(1)\) \(\approx\) \(0.7851419591 - 0.4386656353i\)
\(L(1)\) \(\approx\) \(0.7851419591 - 0.4386656353i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.458 - 0.888i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (0.909 + 0.415i)T \)
17 \( 1 + (-0.690 - 0.723i)T \)
19 \( 1 + (0.723 + 0.690i)T \)
23 \( 1 + (0.945 + 0.327i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (0.995 + 0.0950i)T \)
37 \( 1 + (0.814 + 0.580i)T \)
41 \( 1 + (-0.841 + 0.540i)T \)
43 \( 1 + (-0.989 + 0.142i)T \)
47 \( 1 + (-0.998 - 0.0475i)T \)
53 \( 1 + (-0.945 + 0.327i)T \)
59 \( 1 + (-0.888 + 0.458i)T \)
61 \( 1 + (-0.0475 + 0.998i)T \)
67 \( 1 + (-0.998 + 0.0475i)T \)
71 \( 1 + (-0.959 + 0.281i)T \)
73 \( 1 + (0.189 + 0.981i)T \)
79 \( 1 + (-0.928 + 0.371i)T \)
83 \( 1 + (-0.755 - 0.654i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (-0.989 + 0.142i)T \)
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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

−18.07688326440098950901598897022, −17.45210964758278326784507870378, −17.00307699759460407800508951702, −16.1339749937652158074601981885, −15.7070212806497885704906539162, −15.18446247249364867730755318567, −14.46679072974636449354704188301, −13.47999814032819344108039305029, −13.06161475527366235032286659226, −12.27900606764116316519957258003, −11.537972762821631304625408549151, −10.90196877578331120781385391082, −10.1203837964587939311028881829, −9.23078211919429159208970764231, −8.626037566352296803730575861852, −7.856766021098075854645935897197, −6.81215832300096108544456027733, −6.43765094783461018096394214687, −5.741556965577082527036530068891, −4.86436750994113709180964811872, −4.53471075495475286206998520680, −3.508845885020344453130437324974, −2.955125871328967370006751219895, −1.38093888437259071213474678989, −0.26709006829140195808711015279, 1.151844918415189519204276382930, 1.419173318297913040147783354092, 2.625976843470819936638954869863, 3.27758534783922990194074267415, 4.46088662646526371560002993822, 4.789041886317565391739892178, 5.73280806812177187357407658374, 6.349458082414648141661191484595, 6.98022780685528173941717012040, 8.07293621921382898329051676942, 8.81057923960284627191780167995, 9.77985535183157522566067545542, 10.27260929133904416130347531633, 11.22880327085529031274138907991, 11.53434708301683225346174287861, 12.09106530393714352818012858969, 12.97905653568573007393879328489, 13.53483444788890252951133489448, 13.90943259476146434742660723399, 14.9623082730071699724529343061, 15.73743292380013491980367527667, 16.369155493284620491531950430119, 17.23266093516088767759366399391, 17.99291274631385825673942820305, 18.4236964607456562341095134809

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