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Properties

Label 2-925-185.84-c1-0-35
Degree $2$
Conductor $925$
Sign $0.267 - 0.963i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.267 - 0.963i$
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: \((2,\ 925,\ (\ :1/2),\ 0.267 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.99014 + 2.27259i\)
\(L(\frac12)\) \(\approx\) \(2.99014 + 2.27259i\)
\(L(\frac{3}{2})\) 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

−10.71844871936215791454332339069, −9.267737213663766210472589170953, −8.034314774364274983828794784762, −7.61556034504480481767435950241, −6.67403335535786741824618986419, −5.69984918805793507139769451986, −4.87059234821268480683179464676, −4.40991406475665695658059801704, −3.38710708135790902318687991137, −1.69572164287205558034402985815, 1.52251939900426790271773863869, 2.30248351613197016752542297949, 3.73207969917434607478834238324, 4.42270486597960664898877162095, 5.50596245026313560553045451586, 5.93399795123986813908355216824, 6.97961623049156386644771883563, 8.528616835108446059066821708798, 8.898260403117889308199023807370, 10.39896911568384688903906828130

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