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Properties

Label 2-825-165.38-c1-0-31
Degree $2$
Conductor $825$
Sign $0.108 + 0.994i$
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  + (1.05 − 2.07i)2-s + (−1.42 + 0.980i)3-s + (−2.01 − 2.77i)4-s + (0.526 + 4.00i)6-s + (−0.378 + 2.38i)7-s + (−3.29 + 0.521i)8-s + (1.07 − 2.80i)9-s + (2.56 + 2.10i)11-s + (5.60 + 1.98i)12-s + (0.203 − 0.399i)13-s + (4.56 + 3.31i)14-s + (−0.280 + 0.863i)16-s + (5.33 − 2.72i)17-s + (−4.67 − 5.19i)18-s + (−1.77 + 2.44i)19-s + ⋯
L(s)  = 1  + (0.748 − 1.46i)2-s + (−0.824 + 0.566i)3-s + (−1.00 − 1.38i)4-s + (0.214 + 1.63i)6-s + (−0.143 + 0.902i)7-s + (−1.16 + 0.184i)8-s + (0.358 − 0.933i)9-s + (0.773 + 0.634i)11-s + (1.61 + 0.572i)12-s + (0.0564 − 0.110i)13-s + (1.21 + 0.885i)14-s + (−0.0701 + 0.215i)16-s + (1.29 − 0.659i)17-s + (−1.10 − 1.22i)18-s + (−0.407 + 0.560i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.108 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(825\)    =    \(3 \cdot 5^{2} \cdot 11\)
Sign: $0.108 + 0.994i$
Analytic conductor: \(6.58765\)
Root analytic conductor: \(2.56664\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{825} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 825,\ (\ :1/2),\ 0.108 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33765 - 1.19972i\)
\(L(\frac12)\) \(\approx\) \(1.33765 - 1.19972i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.42 - 0.980i)T \)
5 \( 1 \)
11 \( 1 + (-2.56 - 2.10i)T \)
good2 \( 1 + (-1.05 + 2.07i)T + (-1.17 - 1.61i)T^{2} \)
7 \( 1 + (0.378 - 2.38i)T + (-6.65 - 2.16i)T^{2} \)
13 \( 1 + (-0.203 + 0.399i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-5.33 + 2.72i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (1.77 - 2.44i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (-5.19 + 5.19i)T - 23iT^{2} \)
29 \( 1 + (0.109 - 0.0797i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.851 - 2.62i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-0.349 + 2.20i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-6.72 + 9.25i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (-0.0892 - 0.0892i)T + 43iT^{2} \)
47 \( 1 + (-0.0893 - 0.564i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-7.16 - 3.65i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (10.1 - 7.39i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.745 - 2.29i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.30 + 2.30i)T - 67iT^{2} \)
71 \( 1 + (2.28 + 0.741i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.68 - 0.267i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (-6.06 + 1.97i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.49 - 4.88i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 8.91T + 89T^{2} \)
97 \( 1 + (-3.35 - 1.71i)T + (57.0 + 78.4i)T^{2} \)
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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

−10.37067998079492975285328646220, −9.485662941451652744807902679095, −8.940896604617028863342785160010, −7.25992650756790735909452795375, −6.05292325818363656191370405594, −5.30492057460738490437367167896, −4.48047521693491964802456385708, −3.58123674271769933179252003147, −2.51627563654824866957515977909, −1.05881004303584010930221505885, 1.14859973803409572560523441181, 3.47730561689836899909260413571, 4.42888688978361672734235249236, 5.38364972434183793597298351628, 6.16691481830219639895139756041, 6.78151015980654968852524681159, 7.55268663118351320498644491502, 8.183875742396149071890807891981, 9.412903524598688272977878210976, 10.58827338018039123107240468427

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