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Properties

Label 2-825-11.9-c1-0-4
Degree $2$
Conductor $825$
Sign $-0.836 + 0.548i$
Analytic cond. $6.58765$
Root an. cond. $2.56664$
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  + (−0.575 + 1.77i)2-s + (−0.809 + 0.587i)3-s + (−1.18 − 0.862i)4-s + (−0.575 − 1.77i)6-s + (1.04 + 0.757i)7-s + (−0.802 + 0.583i)8-s + (0.309 − 0.951i)9-s + (3.04 + 1.31i)11-s + 1.46·12-s + (−0.312 + 0.963i)13-s + (−1.94 + 1.41i)14-s + (−1.47 − 4.54i)16-s + (2.19 + 6.76i)17-s + (1.50 + 1.09i)18-s + (−2.69 + 1.95i)19-s + ⋯
L(s)  = 1  + (−0.406 + 1.25i)2-s + (−0.467 + 0.339i)3-s + (−0.593 − 0.431i)4-s + (−0.234 − 0.722i)6-s + (0.394 + 0.286i)7-s + (−0.283 + 0.206i)8-s + (0.103 − 0.317i)9-s + (0.918 + 0.396i)11-s + 0.423·12-s + (−0.0868 + 0.267i)13-s + (−0.518 + 0.376i)14-s + (−0.369 − 1.13i)16-s + (0.532 + 1.64i)17-s + (0.355 + 0.257i)18-s + (−0.617 + 0.448i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $-0.836 + 0.548i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (526, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ -0.836 + 0.548i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.241258 - 0.807337i\)
\(L(\frac12)\) \(\approx\) \(0.241258 - 0.807337i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 \)
11 \( 1 + (-3.04 - 1.31i)T \)
good2 \( 1 + (0.575 - 1.77i)T + (-1.61 - 1.17i)T^{2} \)
7 \( 1 + (-1.04 - 0.757i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.312 - 0.963i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-2.19 - 6.76i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.69 - 1.95i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 4.54T + 23T^{2} \)
29 \( 1 + (-0.510 - 0.370i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.457 + 1.40i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.56 + 1.13i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (8.55 - 6.21i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 1.64T + 43T^{2} \)
47 \( 1 + (-1.61 + 1.17i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-0.544 + 1.67i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.79 - 3.48i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.939 - 2.89i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 10.6T + 67T^{2} \)
71 \( 1 + (-4.99 - 15.3i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.95 + 4.32i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-5.25 + 16.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (2.81 + 8.66i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 0.138T + 89T^{2} \)
97 \( 1 + (-1.66 + 5.13i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.45026758008731155201332986397, −9.788910941682336572108232362942, −8.715181858965427749162331698394, −8.252375987067147632934021832246, −7.22375987114636481036424336023, −6.30299571332037075022054232218, −5.84139956916175568525338704558, −4.71208794347630187682127489145, −3.69805700659324913608111768530, −1.80411400050948049970998868523, 0.51186039812614924273016987211, 1.66139531959615521767980828780, 2.87659135464985297456667718208, 3.98784400191323461727419133838, 5.13705333399379653219705559695, 6.30437218918199179553749809847, 7.12157840262558421332504336241, 8.237801564089393474065913628314, 9.143185423081239014725958012873, 9.912313836159291610150255259146

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