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Properties

Label 2-425-85.49-c1-0-2
Degree $2$
Conductor $425$
Sign $-0.882 + 0.469i$
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  + (−0.187 + 0.187i)2-s + (−1.67 + 0.692i)3-s + 1.92i·4-s + (0.183 − 0.443i)6-s + (−1.88 + 4.55i)7-s + (−0.737 − 0.737i)8-s + (0.193 − 0.193i)9-s + (1.50 − 3.62i)11-s + (−1.33 − 3.22i)12-s + 2.07·13-s + (−0.500 − 1.20i)14-s − 3.58·16-s + (−3.19 − 2.60i)17-s + 0.0724i·18-s + (2.08 + 2.08i)19-s + ⋯
L(s)  = 1  + (−0.132 + 0.132i)2-s + (−0.965 + 0.399i)3-s + 0.964i·4-s + (0.0749 − 0.181i)6-s + (−0.712 + 1.72i)7-s + (−0.260 − 0.260i)8-s + (0.0643 − 0.0643i)9-s + (0.452 − 1.09i)11-s + (−0.385 − 0.931i)12-s + 0.574·13-s + (−0.133 − 0.322i)14-s − 0.895·16-s + (−0.774 − 0.632i)17-s + 0.0170i·18-s + (0.478 + 0.478i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.882 + 0.469i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.882 + 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.100142 - 0.401630i\)
\(L(\frac12)\) \(\approx\) \(0.100142 - 0.401630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
17 \( 1 + (3.19 + 2.60i)T \)
good2 \( 1 + (0.187 - 0.187i)T - 2iT^{2} \)
3 \( 1 + (1.67 - 0.692i)T + (2.12 - 2.12i)T^{2} \)
7 \( 1 + (1.88 - 4.55i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-1.50 + 3.62i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 2.07T + 13T^{2} \)
19 \( 1 + (-2.08 - 2.08i)T + 19iT^{2} \)
23 \( 1 + (3.58 + 1.48i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (3.22 - 1.33i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-2.23 - 5.40i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (1.88 - 0.781i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (10.9 + 4.53i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-3.91 - 3.91i)T + 43iT^{2} \)
47 \( 1 + 0.453T + 47T^{2} \)
53 \( 1 + (-4.60 + 4.60i)T - 53iT^{2} \)
59 \( 1 + (4.92 - 4.92i)T - 59iT^{2} \)
61 \( 1 + (0.0429 + 0.0177i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 10.0iT - 67T^{2} \)
71 \( 1 + (-5.85 - 14.1i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-3.36 - 8.12i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (-0.991 + 2.39i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (1.09 - 1.09i)T - 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + (4.07 + 9.83i)T + (-68.5 + 68.5i)T^{2} \)
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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

−11.77328836774417094073460684148, −11.09079616753636737059764180018, −9.817948420202322783163705261657, −8.747739837325313635656275666768, −8.438596165803879808297845170946, −6.78014164272001814415643440239, −6.01172461258736162366579636241, −5.22063605406197487733516189678, −3.71517948984706654386565611669, −2.64834911852552013712585164348, 0.30982679480224144117546630585, 1.60731589443611713557732608447, 3.80601750776295673275854111932, 4.85316153483298503760138777022, 6.20856395156009613549533778684, 6.61797213634894135149347742101, 7.55066717237777030314148055398, 9.186808307149524051254804571869, 9.969106328917788873385596871282, 10.69226108046517088194086455817

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