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Properties

Label 2-925-185.99-c1-0-9
Degree $2$
Conductor $925$
Sign $-0.751 + 0.659i$
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  + (−0.988 + 0.829i)2-s + (−1.05 + 1.25i)3-s + (−0.0582 + 0.330i)4-s − 2.11i·6-s + (0.976 + 2.68i)7-s + (−1.50 − 2.60i)8-s + (0.0552 + 0.313i)9-s + (1.17 + 2.03i)11-s + (−0.353 − 0.420i)12-s + (−0.449 + 2.55i)13-s + (−3.18 − 1.84i)14-s + (3.02 + 1.10i)16-s + (1.16 + 6.62i)17-s + (−0.314 − 0.263i)18-s + (4.53 − 5.39i)19-s + ⋯
L(s)  = 1  + (−0.698 + 0.586i)2-s + (−0.607 + 0.724i)3-s + (−0.0291 + 0.165i)4-s − 0.862i·6-s + (0.368 + 1.01i)7-s + (−0.532 − 0.922i)8-s + (0.0184 + 0.104i)9-s + (0.354 + 0.614i)11-s + (−0.101 − 0.121i)12-s + (−0.124 + 0.707i)13-s + (−0.852 − 0.492i)14-s + (0.755 + 0.275i)16-s + (0.283 + 1.60i)17-s + (−0.0741 − 0.0622i)18-s + (1.03 − 1.23i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.751 + 0.659i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.751 + 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.256262 - 0.681090i\)
\(L(\frac12)\) \(\approx\) \(0.256262 - 0.681090i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.46 + 5.55i)T \)
good2 \( 1 + (0.988 - 0.829i)T + (0.347 - 1.96i)T^{2} \)
3 \( 1 + (1.05 - 1.25i)T + (-0.520 - 2.95i)T^{2} \)
7 \( 1 + (-0.976 - 2.68i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (-1.17 - 2.03i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.449 - 2.55i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (-1.16 - 6.62i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-4.53 + 5.39i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (4.47 - 7.75i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.496 - 0.286i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.07iT - 31T^{2} \)
41 \( 1 + (1.54 - 8.74i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 1.85T + 43T^{2} \)
47 \( 1 + (-4.32 - 2.49i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.67 + 12.8i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (-0.214 + 0.588i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (2.25 + 0.397i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (4.21 + 11.5i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (2.29 + 1.92i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 12.2iT - 73T^{2} \)
79 \( 1 + (0.188 + 0.518i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.71 - 0.303i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-2.18 + 6.00i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-1.67 + 2.90i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.35395826931042786460583765008, −9.511967192447035409858576266677, −9.082465262405615919059041474294, −8.064413577692943293355111818916, −7.37494881237866812442632622153, −6.29372439448922925379216394435, −5.48827199536527749106004304442, −4.52505079669768247201591382447, −3.50702630734159470620594580585, −1.85830628826534557336607209510, 0.55509561260123313446479177198, 1.18447091922464895143271801530, 2.69911546045136466090611058163, 4.05997560624765573784194922857, 5.41786732754947638591922304786, 6.04248569994135869513643226208, 7.18506619089544784692255916330, 7.83527327660654479317111684108, 8.815774686945247874571186137742, 9.825973416192746670968410797337

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