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Properties

Label 2-925-185.184-c1-0-49
Degree $2$
Conductor $925$
Sign $-0.560 - 0.828i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.36·2-s − 2.79i·3-s − 0.135·4-s + 3.81i·6-s − 4.67i·7-s + 2.91·8-s − 4.80·9-s − 0.186·11-s + 0.378i·12-s − 4.24·13-s + 6.38i·14-s − 3.71·16-s + 3.61·17-s + 6.55·18-s − 7.92i·19-s + ⋯
L(s)  = 1  − 0.965·2-s − 1.61i·3-s − 0.0676·4-s + 1.55i·6-s − 1.76i·7-s + 1.03·8-s − 1.60·9-s − 0.0561·11-s + 0.109i·12-s − 1.17·13-s + 1.70i·14-s − 0.927·16-s + 0.876·17-s + 1.54·18-s − 1.81i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.560 - 0.828i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.560 - 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.560 - 0.828i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (924, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.560 - 0.828i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.242081 + 0.456162i\)
\(L(\frac12)\) \(\approx\) \(0.242081 + 0.456162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 + (-2.98 + 5.30i)T \)
good2 \( 1 + 1.36T + 2T^{2} \)
3 \( 1 + 2.79iT - 3T^{2} \)
7 \( 1 + 4.67iT - 7T^{2} \)
11 \( 1 + 0.186T + 11T^{2} \)
13 \( 1 + 4.24T + 13T^{2} \)
17 \( 1 - 3.61T + 17T^{2} \)
19 \( 1 + 7.92iT - 19T^{2} \)
23 \( 1 + 3.32T + 23T^{2} \)
29 \( 1 - 2.01iT - 29T^{2} \)
31 \( 1 - 2.04iT - 31T^{2} \)
41 \( 1 - 1.03T + 41T^{2} \)
43 \( 1 - 4.05T + 43T^{2} \)
47 \( 1 + 2.78iT - 47T^{2} \)
53 \( 1 + 7.67iT - 53T^{2} \)
59 \( 1 - 9.12iT - 59T^{2} \)
61 \( 1 - 12.5iT - 61T^{2} \)
67 \( 1 - 7.35iT - 67T^{2} \)
71 \( 1 - 15.2T + 71T^{2} \)
73 \( 1 - 0.622iT - 73T^{2} \)
79 \( 1 + 0.662iT - 79T^{2} \)
83 \( 1 - 9.75iT - 83T^{2} \)
89 \( 1 + 6.06iT - 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.509221541416556036463554316984, −8.485754148473105013991303710191, −7.53909178940313152902696658323, −7.36538700522430386500875818372, −6.68442878543735760351702427130, −5.18711913685728908530046049149, −4.09090864170131512098075990427, −2.48759265347357708445538438860, −1.16166101905361229618224704848, −0.39134263138088370214151331239, 2.11782889813845657706558174891, 3.38692685325110031456814818793, 4.53281925782908034623272866431, 5.31009400770580291584891457593, 6.04408423367930018471793028644, 7.86862951966688605738349949146, 8.265214683858826241881874285035, 9.346561697751758947264638512130, 9.632122295360196303704872211132, 10.15864859053361517980049146309

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