Properties

Label 2-925-37.3-c1-0-9
Degree $2$
Conductor $925$
Sign $-0.957 + 0.289i$
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  + (−1.29 + 1.53i)2-s + (−0.583 + 0.489i)3-s + (−0.353 − 2.00i)4-s − 1.53i·6-s + (1.63 + 0.593i)7-s + (0.0651 + 0.0376i)8-s + (−0.420 + 2.38i)9-s + (−0.381 + 0.661i)11-s + (1.18 + 0.998i)12-s + (0.0386 − 0.00681i)13-s + (−3.02 + 1.74i)14-s + (3.68 − 1.34i)16-s + (4.30 + 0.758i)17-s + (−3.12 − 3.72i)18-s + (2.89 + 3.45i)19-s + ⋯
L(s)  = 1  + (−0.913 + 1.08i)2-s + (−0.337 + 0.282i)3-s + (−0.176 − 1.00i)4-s − 0.625i·6-s + (0.616 + 0.224i)7-s + (0.0230 + 0.0132i)8-s + (−0.140 + 0.794i)9-s + (−0.115 + 0.199i)11-s + (0.343 + 0.288i)12-s + (0.0107 − 0.00188i)13-s + (−0.807 + 0.466i)14-s + (0.921 − 0.335i)16-s + (1.04 + 0.184i)17-s + (−0.736 − 0.877i)18-s + (0.664 + 0.791i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0980587 - 0.661962i\)
\(L(\frac12)\) \(\approx\) \(0.0980587 - 0.661962i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 - 5.76i)T \)
good2 \( 1 + (1.29 - 1.53i)T + (-0.347 - 1.96i)T^{2} \)
3 \( 1 + (0.583 - 0.489i)T + (0.520 - 2.95i)T^{2} \)
7 \( 1 + (-1.63 - 0.593i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (0.381 - 0.661i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0386 + 0.00681i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-4.30 - 0.758i)T + (15.9 + 5.81i)T^{2} \)
19 \( 1 + (-2.89 - 3.45i)T + (-3.29 + 18.7i)T^{2} \)
23 \( 1 + (1.58 - 0.913i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.07 + 2.35i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.00iT - 31T^{2} \)
41 \( 1 + (-1.01 - 5.75i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + 3.43iT - 43T^{2} \)
47 \( 1 + (-1.81 - 3.14i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.2 - 4.08i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (2.59 + 7.12i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (5.71 - 1.00i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (0.268 + 0.0977i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (4.33 - 3.64i)T + (12.3 - 69.9i)T^{2} \)
73 \( 1 - 4.13T + 73T^{2} \)
79 \( 1 + (0.799 - 2.19i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.460 - 2.61i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (5.87 + 16.1i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-12.8 + 7.43i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.15749847944023157306427676307, −9.710398213550902930499618311902, −8.625587873580902653064308559035, −7.86514824307324059067469824454, −7.53433525041835541836611350345, −6.24171523150602499417776717950, −5.54945505484033277260445160041, −4.75154966525915515640483873696, −3.25224649870287643001121506927, −1.52974873435615557634339874905, 0.48375493979553696852658877215, 1.53948631248813396973217558177, 2.84851492591409224375895609366, 3.83912529692288099408624080454, 5.28392884185594861591289092442, 6.13334472850012056217372448659, 7.40020485472285396252192553004, 8.043375842950447936787876916480, 9.182388678789535504898151345451, 9.511427885293115952823173985163

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