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Properties

Label 2-925-185.174-c1-0-22
Degree $2$
Conductor $925$
Sign $-0.326 - 0.945i$
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  + (−0.978 + 0.564i)2-s + (2.45 + 1.41i)3-s + (−0.361 + 0.626i)4-s − 3.20·6-s + (−0.203 − 0.117i)7-s − 3.07i·8-s + (2.51 + 4.35i)9-s + 3.97·11-s + (−1.77 + 1.02i)12-s + (5.09 + 2.94i)13-s + 0.265·14-s + (1.01 + 1.75i)16-s + (−2.17 + 1.25i)17-s + (−4.91 − 2.83i)18-s + (2.90 − 5.03i)19-s + ⋯
L(s)  = 1  + (−0.691 + 0.399i)2-s + (1.41 + 0.817i)3-s + (−0.180 + 0.313i)4-s − 1.30·6-s + (−0.0767 − 0.0443i)7-s − 1.08i·8-s + (0.837 + 1.45i)9-s + 1.19·11-s + (−0.512 + 0.295i)12-s + (1.41 + 0.816i)13-s + 0.0708·14-s + (0.253 + 0.439i)16-s + (−0.527 + 0.304i)17-s + (−1.15 − 0.669i)18-s + (0.666 − 1.15i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.326 - 0.945i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05016 + 1.47460i\)
\(L(\frac12)\) \(\approx\) \(1.05016 + 1.47460i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.100 - 6.08i)T \)
good2 \( 1 + (0.978 - 0.564i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (-2.45 - 1.41i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.203 + 0.117i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.97T + 11T^{2} \)
13 \( 1 + (-5.09 - 2.94i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.17 - 1.25i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.90 + 5.03i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.07iT - 23T^{2} \)
29 \( 1 + 1.36T + 29T^{2} \)
31 \( 1 + 4.89T + 31T^{2} \)
41 \( 1 + (3.13 - 5.42i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 4.19iT - 43T^{2} \)
47 \( 1 + 2.34iT - 47T^{2} \)
53 \( 1 + (9.19 - 5.31i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.61 + 6.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.77 + 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.644 + 0.372i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.29 + 2.24i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 11.7iT - 73T^{2} \)
79 \( 1 + (-4.73 + 8.20i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.0157 + 0.00910i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.76 + 6.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.0iT - 97T^{2} \)
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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

−9.642929107625320181708236086619, −9.408537534577419524659437526838, −8.720706189810157958249152013076, −8.206727255411339620637445956594, −7.14564460394292330999808513173, −6.41245795640165706905848013954, −4.69801347356917142658366096485, −3.74659537665330014255844485100, −3.34844492745339880541463750309, −1.61577596808356899853319247757, 1.07330918515852870855678077757, 1.89779572777127373987266859848, 3.12337738278206288411355843572, 4.04863173495736827410124499376, 5.67741379886319490508738607142, 6.56157294952478036109961514059, 7.66690901394808819678584204460, 8.358729062059129075667309627386, 8.978608481994008786404059397143, 9.471230562170900823600154324628

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