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Properties

Label 2-925-37.10-c1-0-5
Degree $2$
Conductor $925$
Sign $-0.264 + 0.964i$
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  + (−0.945 + 1.63i)2-s + (1.39 + 2.42i)3-s + (−0.786 − 1.36i)4-s − 5.28·6-s + (1.45 + 2.51i)7-s − 0.807·8-s + (−2.40 + 4.16i)9-s − 1.47·11-s + (2.19 − 3.80i)12-s + (−2.69 − 4.66i)13-s − 5.49·14-s + (2.33 − 4.04i)16-s + (−2.56 + 4.44i)17-s + (−4.54 − 7.87i)18-s + (−0.298 − 0.517i)19-s + ⋯
L(s)  = 1  + (−0.668 + 1.15i)2-s + (0.806 + 1.39i)3-s + (−0.393 − 0.681i)4-s − 2.15·6-s + (0.549 + 0.951i)7-s − 0.285·8-s + (−0.801 + 1.38i)9-s − 0.444·11-s + (0.634 − 1.09i)12-s + (−0.746 − 1.29i)13-s − 1.46·14-s + (0.583 − 1.01i)16-s + (−0.621 + 1.07i)17-s + (−1.07 − 1.85i)18-s + (−0.0685 − 0.118i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.264 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.264 + 0.964i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.264 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.625675 - 0.820185i\)
\(L(\frac12)\) \(\approx\) \(0.625675 - 0.820185i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.45 - 5.56i)T \)
good2 \( 1 + (0.945 - 1.63i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.39 - 2.42i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (-1.45 - 2.51i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.47T + 11T^{2} \)
13 \( 1 + (2.69 + 4.66i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.56 - 4.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.298 + 0.517i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.262T + 23T^{2} \)
29 \( 1 - 4.49T + 29T^{2} \)
31 \( 1 + 5.64T + 31T^{2} \)
41 \( 1 + (-4.97 - 8.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 9.62T + 43T^{2} \)
47 \( 1 + 1.89T + 47T^{2} \)
53 \( 1 + (-3.68 + 6.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (7.30 - 12.6i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.78 + 6.54i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.570 + 0.988i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.83 - 11.8i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.7T + 73T^{2} \)
79 \( 1 + (-5.86 - 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.15 + 3.72i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.73 - 6.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−10.20943957911709811777809036870, −9.663320184006934330110158232415, −8.698546486453068617093956504545, −8.362520669700851155940368356613, −7.71272606096241839735220764633, −6.40674274024294637077040248691, −5.35987764670724178521896878748, −4.82941723506172814332034977441, −3.38842024752651502576990743180, −2.47642706912371924051148887888, 0.52047249431979915256999076323, 1.79629710427470336352219098711, 2.35123136537484728156379644091, 3.52595496299006883789912688236, 4.80737249870765866274472868182, 6.45145394137582767120521560133, 7.25589105399867966030874786039, 7.79712173551076254668752187554, 8.879232432077738380314568471693, 9.300693395746672897637681611513

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