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Properties

Label 2-925-185.84-c1-0-3
Degree $2$
Conductor $925$
Sign $-0.157 - 0.987i$
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  + (−0.479 − 0.276i)2-s + (1.80 − 1.04i)3-s + (−0.846 − 1.46i)4-s − 1.15·6-s + (−3.61 + 2.08i)7-s + 2.04i·8-s + (0.670 − 1.16i)9-s − 3.31·11-s + (−3.05 − 1.76i)12-s + (0.365 − 0.211i)13-s + 2.30·14-s + (−1.12 + 1.95i)16-s + (1.57 + 0.909i)17-s + (−0.643 + 0.371i)18-s + (1.28 + 2.22i)19-s + ⋯
L(s)  = 1  + (−0.339 − 0.195i)2-s + (1.04 − 0.601i)3-s + (−0.423 − 0.733i)4-s − 0.470·6-s + (−1.36 + 0.787i)7-s + 0.723i·8-s + (0.223 − 0.387i)9-s − 0.998·11-s + (−0.882 − 0.509i)12-s + (0.101 − 0.0585i)13-s + 0.617·14-s + (−0.281 + 0.488i)16-s + (0.381 + 0.220i)17-s + (−0.151 + 0.0875i)18-s + (0.294 + 0.509i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.157 - 0.987i$
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: \((2,\ 925,\ (\ :1/2),\ -0.157 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.238464 + 0.279420i\)
\(L(\frac12)\) \(\approx\) \(0.238464 + 0.279420i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.66 - 5.46i)T \)
good2 \( 1 + (0.479 + 0.276i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.80 + 1.04i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (3.61 - 2.08i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.31T + 11T^{2} \)
13 \( 1 + (-0.365 + 0.211i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.57 - 0.909i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.28 - 2.22i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.981iT - 23T^{2} \)
29 \( 1 + 7.35T + 29T^{2} \)
31 \( 1 + 6.58T + 31T^{2} \)
41 \( 1 + (3.14 + 5.44i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 8.10iT - 43T^{2} \)
47 \( 1 - 10.6iT - 47T^{2} \)
53 \( 1 + (-5.40 - 3.11i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.556 - 0.963i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.40 - 5.90i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.05 + 3.49i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.43 + 11.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.36iT - 73T^{2} \)
79 \( 1 + (7.18 + 12.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.959 + 0.554i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.18 - 5.50i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.47iT - 97T^{2} \)
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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

−10.10657390635089228805315399918, −9.304300754557608371065550689128, −8.832031488202566374555790762710, −7.955266773359867432405495064892, −7.10293965456295584543476937309, −5.85304915543130885165741942818, −5.36455051956463464705775079223, −3.63625242701572577462005391555, −2.71470452627382774452607111870, −1.74883132237699613390314171063, 0.16101956856311775298553926254, 2.69953790747482058764446073518, 3.48980744399719732600908019311, 4.03233623572931738813064768659, 5.36951076106049531324336510404, 6.79578607353195953300938626671, 7.45637020184722785467928450172, 8.267563122051882203965484426498, 9.097943651014831778897196255519, 9.690149063449980016215700226227

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