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Properties

Label 2-925-185.174-c1-0-7
Degree $2$
Conductor $925$
Sign $-0.0679 - 0.997i$
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  + (1.06 − 0.613i)2-s + (−0.819 − 0.473i)3-s + (−0.246 + 0.426i)4-s − 1.16·6-s + (−1.74 − 1.00i)7-s + 3.06i·8-s + (−1.05 − 1.82i)9-s − 0.974·11-s + (0.403 − 0.233i)12-s + (4.65 + 2.68i)13-s − 2.47·14-s + (1.38 + 2.40i)16-s + (−6.57 + 3.79i)17-s + (−2.23 − 1.29i)18-s + (0.489 − 0.847i)19-s + ⋯
L(s)  = 1  + (0.751 − 0.434i)2-s + (−0.473 − 0.273i)3-s + (−0.123 + 0.213i)4-s − 0.474·6-s + (−0.660 − 0.381i)7-s + 1.08i·8-s + (−0.350 − 0.607i)9-s − 0.293·11-s + (0.116 − 0.0673i)12-s + (1.29 + 0.745i)13-s − 0.662·14-s + (0.346 + 0.600i)16-s + (−1.59 + 0.920i)17-s + (−0.527 − 0.304i)18-s + (0.112 − 0.194i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0679 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0679 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.0679 - 0.997i$
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: \((2,\ 925,\ (\ :1/2),\ -0.0679 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.591726 + 0.633402i\)
\(L(\frac12)\) \(\approx\) \(0.591726 + 0.633402i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.81 - 4.73i)T \)
good2 \( 1 + (-1.06 + 0.613i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.819 + 0.473i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.74 + 1.00i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.974T + 11T^{2} \)
13 \( 1 + (-4.65 - 2.68i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (6.57 - 3.79i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.489 + 0.847i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 7.57iT - 23T^{2} \)
29 \( 1 + 3.68T + 29T^{2} \)
31 \( 1 + 1.02T + 31T^{2} \)
41 \( 1 + (4.57 - 7.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 4.36iT - 43T^{2} \)
47 \( 1 + 8.32iT - 47T^{2} \)
53 \( 1 + (10.9 - 6.31i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.00 - 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.168 - 0.291i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.817 + 0.471i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.32 + 9.21i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 + (-8.73 + 15.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.992 + 0.573i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.34 + 2.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.0iT - 97T^{2} \)
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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

−10.64563212016314807605921385424, −9.342027608296794223848766874418, −8.757718569914412830281120921860, −7.72556485476649344007235871753, −6.51169927344159185262445846767, −6.08279495857339536578724957390, −4.87873314707504176185319392271, −3.84340632225400753822967299272, −3.25404145477188837430148827610, −1.70907838946464303226930606489, 0.32989215120061601843080074213, 2.48072729440401059717699126411, 3.74612141616389866946536532602, 4.73833581473335453770470273508, 5.49994205602311232832453995035, 6.19326526116457652892039753362, 6.90091654751204930597309528926, 8.212285932067102426241142410338, 9.042327719031945926599173020075, 9.922861835879431632759459617140

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