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Properties

Label 2-925-185.162-c1-0-7
Degree $2$
Conductor $925$
Sign $0.239 - 0.970i$
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  + (0.212 + 0.122i)2-s + (0.491 − 1.83i)3-s + (−0.969 − 1.67i)4-s + (0.329 − 0.329i)6-s + (−1.00 + 3.76i)7-s − 0.967i·8-s + (−0.527 − 0.304i)9-s + 4.37i·11-s + (−3.55 + 0.953i)12-s + (−4.43 + 2.55i)13-s + (−0.676 + 0.676i)14-s + (−1.82 + 3.15i)16-s + (−1.24 + 2.15i)17-s + (−0.0746 − 0.129i)18-s + (−0.196 + 0.731i)19-s + ⋯
L(s)  = 1  + (0.150 + 0.0867i)2-s + (0.283 − 1.05i)3-s + (−0.484 − 0.839i)4-s + (0.134 − 0.134i)6-s + (−0.381 + 1.42i)7-s − 0.341i·8-s + (−0.175 − 0.101i)9-s + 1.31i·11-s + (−1.02 + 0.275i)12-s + (−1.22 + 0.709i)13-s + (−0.180 + 0.180i)14-s + (−0.455 + 0.788i)16-s + (−0.301 + 0.521i)17-s + (−0.0176 − 0.0304i)18-s + (−0.0449 + 0.167i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.239 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.239 - 0.970i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (532, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.239 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670476 + 0.525333i\)
\(L(\frac12)\) \(\approx\) \(0.670476 + 0.525333i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.11 + 5.22i)T \)
good2 \( 1 + (-0.212 - 0.122i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.491 + 1.83i)T + (-2.59 - 1.5i)T^{2} \)
7 \( 1 + (1.00 - 3.76i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 - 4.37iT - 11T^{2} \)
13 \( 1 + (4.43 - 2.55i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.24 - 2.15i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.196 - 0.731i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 3.38iT - 23T^{2} \)
29 \( 1 + (4.99 - 4.99i)T - 29iT^{2} \)
31 \( 1 + (5.96 + 5.96i)T + 31iT^{2} \)
41 \( 1 + (-4.18 + 2.41i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + 3.40iT - 43T^{2} \)
47 \( 1 + (5.91 + 5.91i)T + 47iT^{2} \)
53 \( 1 + (-0.158 - 0.590i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (9.92 - 2.66i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.63 - 9.83i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-12.4 - 3.32i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-5.23 - 9.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.10 - 3.10i)T + 73iT^{2} \)
79 \( 1 + (0.743 - 2.77i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (1.55 + 5.80i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (0.387 + 1.44i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + 6.03T + 97T^{2} \)
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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

−9.900254766547506871120479955367, −9.432911254849352356963231510030, −8.724583756810501273505856679661, −7.48006476601273657161003560631, −6.93863422420555210101592698155, −5.91476700383657376832095556808, −5.18258255435140330576937162157, −4.11482135025843437226419847826, −2.27807788333323401344417088968, −1.83004258952700362744231117651, 0.35534030902211012670968194690, 2.95254699125339626100669625622, 3.52304993378591987671940495839, 4.41987001900302465460146363315, 5.07288686645186498468142896315, 6.56543312794555760928413221369, 7.56757704639796663896475613797, 8.200378638364462290818884729544, 9.339692488975635876768827649529, 9.709930709418030763386880410503

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