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Properties

Label 2-925-185.117-c1-0-32
Degree $2$
Conductor $925$
Sign $-0.0158 + 0.999i$
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-s + (−1 − i)3-s − 4-s + (−1 − i)6-s + (3 + 3i)7-s − 3·8-s − i·9-s − 2i·11-s + (1 + i)12-s − 2·13-s + (3 + 3i)14-s − 16-s − 4i·17-s − i·18-s + (3 − 3i)19-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.577 − 0.577i)3-s − 0.5·4-s + (−0.408 − 0.408i)6-s + (1.13 + 1.13i)7-s − 1.06·8-s − 0.333i·9-s − 0.603i·11-s + (0.288 + 0.288i)12-s − 0.554·13-s + (0.801 + 0.801i)14-s − 0.250·16-s − 0.970i·17-s − 0.235i·18-s + (0.688 − 0.688i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0158 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0158 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.0158 + 0.999i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.0158 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.994103 - 1.01003i\)
\(L(\frac12)\) \(\approx\) \(0.994103 - 1.01003i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 6i)T \)
good2 \( 1 - T + 2T^{2} \)
3 \( 1 + (1 + i)T + 3iT^{2} \)
7 \( 1 + (-3 - 3i)T + 7iT^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + (-3 + 3i)T - 19iT^{2} \)
23 \( 1 - 8T + 23T^{2} \)
29 \( 1 + (7 + 7i)T + 29iT^{2} \)
31 \( 1 + (-3 + 3i)T - 31iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (5 + 5i)T + 47iT^{2} \)
53 \( 1 + (-3 + 3i)T - 53iT^{2} \)
59 \( 1 + (-7 + 7i)T - 59iT^{2} \)
61 \( 1 + (-1 + i)T - 61iT^{2} \)
67 \( 1 + (3 - 3i)T - 67iT^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (1 + i)T + 73iT^{2} \)
79 \( 1 + (-3 + 3i)T - 79iT^{2} \)
83 \( 1 + (-5 + 5i)T - 83iT^{2} \)
89 \( 1 + (-5 - 5i)T + 89iT^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−9.585596485488115110706128554659, −9.092096962415640318575093717439, −8.198988079779502516560133945599, −7.17555806445055832104720949487, −6.16533800907379519059950500906, −5.19772248993176353649746483620, −5.03574541911461831534542273599, −3.51225529781775833170540860983, −2.35822040831558650167755421969, −0.62730601236028749827165058491, 1.46757420986876449562073027714, 3.34474249918042278093347192068, 4.33800225605344126432721724490, 4.93441539089070830944985017230, 5.43443841568821475809739782313, 6.82110386031872740768304699598, 7.72421514620521137727135521916, 8.562738078495050675934083390182, 9.716834704066526040423319756452, 10.36064411301665304608862811310

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