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Properties

Label 2-925-185.23-c1-0-53
Degree $2$
Conductor $925$
Sign $-0.0382 + 0.999i$
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  + (1.14 + 0.659i)2-s + (1.18 + 0.318i)3-s + (−0.131 − 0.227i)4-s + (1.14 + 1.14i)6-s + (−4.13 − 1.10i)7-s − 2.98i·8-s + (−1.28 − 0.743i)9-s − 4.89i·11-s + (−0.0835 − 0.311i)12-s + (−2.57 + 1.48i)13-s + (−3.98 − 3.98i)14-s + (1.70 − 2.94i)16-s + (−2.10 + 3.65i)17-s + (−0.979 − 1.69i)18-s + (4.90 + 1.31i)19-s + ⋯
L(s)  = 1  + (0.807 + 0.466i)2-s + (0.686 + 0.183i)3-s + (−0.0656 − 0.113i)4-s + (0.468 + 0.468i)6-s + (−1.56 − 0.418i)7-s − 1.05i·8-s + (−0.429 − 0.247i)9-s − 1.47i·11-s + (−0.0241 − 0.0900i)12-s + (−0.714 + 0.412i)13-s + (−1.06 − 1.06i)14-s + (0.425 − 0.737i)16-s + (−0.511 + 0.885i)17-s + (−0.230 − 0.400i)18-s + (1.12 + 0.301i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0382 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0382 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.0382 + 0.999i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.0382 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08938 - 1.13191i\)
\(L(\frac12)\) \(\approx\) \(1.08938 - 1.13191i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.219 - 6.07i)T \)
good2 \( 1 + (-1.14 - 0.659i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-1.18 - 0.318i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (4.13 + 1.10i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + 4.89iT - 11T^{2} \)
13 \( 1 + (2.57 - 1.48i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.10 - 3.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.90 - 1.31i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 6.57iT - 23T^{2} \)
29 \( 1 + (0.417 + 0.417i)T + 29iT^{2} \)
31 \( 1 + (-3.03 + 3.03i)T - 31iT^{2} \)
41 \( 1 + (0.897 - 0.518i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 - 3.17iT - 43T^{2} \)
47 \( 1 + (-4.44 + 4.44i)T - 47iT^{2} \)
53 \( 1 + (-6.40 + 1.71i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.57 - 5.86i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.68 - 0.450i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-3.09 + 11.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.78 + 3.09i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.50 - 2.50i)T - 73iT^{2} \)
79 \( 1 + (3.97 + 1.06i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (2.57 - 0.688i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (11.4 - 3.07i)T + (77.0 - 44.5i)T^{2} \)
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

−9.811889982687942515940535349715, −9.061029733154763010955178216879, −8.266761981093806751384555168874, −6.99260647262794088895816174369, −6.26915501395976702304195720235, −5.71692844016436122575351332457, −4.34305596813934070388636760440, −3.52155899523819521338135075734, −2.84780663046855926597614262081, −0.49157562990612398250468197415, 2.31041874781763945541830426050, 2.87432905241614824244813799356, 3.71899417401182827383751208507, 4.96599422866294393766485337372, 5.65148309453756025141031022187, 7.12481479115031505801593626880, 7.54105770251164376657395918404, 8.846663923886642074950807721944, 9.426883805200626759425508102037, 10.11094578428743061706059844734

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