Properties

Label 2-425-425.84-c1-0-17
Degree $2$
Conductor $425$
Sign $0.834 - 0.550i$
Analytic cond. $3.39364$
Root an. cond. $1.84218$
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  + (−1.39 + 1.91i)2-s + (0.152 + 0.469i)3-s + (−1.12 − 3.45i)4-s + (−1.97 + 1.04i)5-s + (−1.11 − 0.362i)6-s + 1.83·7-s + (3.67 + 1.19i)8-s + (2.22 − 1.62i)9-s + (0.756 − 5.25i)10-s + (3.32 − 4.58i)11-s + (1.45 − 1.05i)12-s + (−3.63 − 5.00i)13-s + (−2.56 + 3.52i)14-s + (−0.791 − 0.769i)15-s + (−1.54 + 1.12i)16-s + (−3.84 + 1.47i)17-s + ⋯
L(s)  = 1  + (−0.986 + 1.35i)2-s + (0.0881 + 0.271i)3-s + (−0.560 − 1.72i)4-s + (−0.884 + 0.466i)5-s + (−0.454 − 0.147i)6-s + 0.694·7-s + (1.29 + 0.422i)8-s + (0.743 − 0.540i)9-s + (0.239 − 1.66i)10-s + (1.00 − 1.38i)11-s + (0.418 − 0.304i)12-s + (−1.00 − 1.38i)13-s + (−0.684 + 0.942i)14-s + (−0.204 − 0.198i)15-s + (−0.387 + 0.281i)16-s + (−0.933 + 0.358i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.834 - 0.550i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.834 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.683432 + 0.205262i\)
\(L(\frac12)\) \(\approx\) \(0.683432 + 0.205262i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.97 - 1.04i)T \)
17 \( 1 + (3.84 - 1.47i)T \)
good2 \( 1 + (1.39 - 1.91i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (-0.152 - 0.469i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 - 1.83T + 7T^{2} \)
11 \( 1 + (-3.32 + 4.58i)T + (-3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.63 + 5.00i)T + (-4.01 + 12.3i)T^{2} \)
19 \( 1 + (-0.857 + 2.63i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.77 - 2.01i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (4.00 - 1.30i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.14 - 1.02i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.35 + 1.71i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-0.870 - 1.19i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 1.83iT - 43T^{2} \)
47 \( 1 + (-4.47 + 1.45i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-8.15 + 2.64i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.99 + 3.62i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.602 + 0.829i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.30 + 1.07i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (12.9 - 4.20i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (10.7 + 7.80i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-10.2 + 3.31i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (8.43 + 2.74i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + (-2.02 - 1.47i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.152 + 0.467i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.00305664821976118355555513614, −10.16771665982847867154737810888, −9.070645898819204944340949035766, −8.477986219852071171805344542874, −7.53129459775134656617364064156, −6.92666366877939980059204834910, −5.87475387812403570866716616099, −4.66295954008364418936888359247, −3.33630626517221428268695327407, −0.71022710625544169334280262468, 1.42649604534487948089453398233, 2.28559585258441116667835139666, 4.24353959268099130970510431343, 4.53761672530695720796143792922, 7.04450553994296832855811473773, 7.55434196395980954470377197726, 8.669668809590564506720280752051, 9.366889529279183520036687263226, 10.13360438094886002702020604883, 11.28552289904828046600212548971

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