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Properties

Label 2-925-185.84-c1-0-26
Degree $2$
Conductor $925$
Sign $-0.931 + 0.363i$
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  + (−2.06 − 1.19i)2-s + (0.547 − 0.316i)3-s + (1.84 + 3.20i)4-s − 1.50·6-s + (−4.45 + 2.57i)7-s − 4.05i·8-s + (−1.30 + 2.25i)9-s + 2.91·11-s + (2.02 + 1.16i)12-s + (−1.55 + 0.899i)13-s + 12.2·14-s + (−1.14 + 1.98i)16-s + (2.02 + 1.16i)17-s + (5.37 − 3.10i)18-s + (−1.55 − 2.68i)19-s + ⋯
L(s)  = 1  + (−1.46 − 0.844i)2-s + (0.316 − 0.182i)3-s + (0.924 + 1.60i)4-s − 0.616·6-s + (−1.68 + 0.971i)7-s − 1.43i·8-s + (−0.433 + 0.750i)9-s + 0.878·11-s + (0.584 + 0.337i)12-s + (−0.431 + 0.249i)13-s + 3.28·14-s + (−0.285 + 0.495i)16-s + (0.490 + 0.282i)17-s + (1.26 − 0.731i)18-s + (−0.356 − 0.617i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.931 + 0.363i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.931 + 0.363i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0415623 - 0.220578i\)
\(L(\frac12)\) \(\approx\) \(0.0415623 - 0.220578i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.69 + 3.86i)T \)
good2 \( 1 + (2.06 + 1.19i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.547 + 0.316i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (4.45 - 2.57i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.91T + 11T^{2} \)
13 \( 1 + (1.55 - 0.899i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.02 - 1.16i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.55 + 2.68i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.53iT - 23T^{2} \)
29 \( 1 + 6.98T + 29T^{2} \)
31 \( 1 + 1.47T + 31T^{2} \)
41 \( 1 + (-0.463 - 0.803i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 8.91iT - 43T^{2} \)
47 \( 1 + 8.87iT - 47T^{2} \)
53 \( 1 + (-4.38 - 2.53i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.338 - 0.586i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.43 + 11.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.20 - 1.84i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.775 + 1.34i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 14.1iT - 73T^{2} \)
79 \( 1 + (2.89 + 5.01i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.32 - 4.80i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.54 - 6.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 18.3iT - 97T^{2} \)
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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

−9.560605729497492065126668951182, −8.989295311512876100200829550499, −8.509744696475036322304100185476, −7.38028237483360126143491998085, −6.59297278403263227456670746744, −5.52124286105799187962385880074, −3.74036903766865444799129839920, −2.71364023440140763238450390490, −2.07488950549704997109959425895, −0.18380007729168057303953798715, 1.13845216259960897860388550481, 3.15285920947080580929041179976, 3.96062328888978671479811636172, 5.89707231852611871665660379107, 6.34681633027419112593396918916, 7.26765522493962293283478517634, 7.80540038000709938626620079539, 9.060483707257895536386174848224, 9.513297578973037579961097561192, 9.873594283113736743230452717445

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