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Properties

Label 2-425-85.4-c1-0-16
Degree $2$
Conductor $425$
Sign $0.869 - 0.494i$
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  + 2.08·2-s + (1.31 + 1.31i)3-s + 2.34·4-s + (2.73 + 2.73i)6-s + (0.971 − 0.971i)7-s + 0.713·8-s + 0.435i·9-s + (−0.384 − 0.384i)11-s + (3.07 + 3.07i)12-s + 5.39i·13-s + (2.02 − 2.02i)14-s − 3.19·16-s + (−3.38 − 2.36i)17-s + 0.907i·18-s − 4.86i·19-s + ⋯
L(s)  = 1  + 1.47·2-s + (0.756 + 0.756i)3-s + 1.17·4-s + (1.11 + 1.11i)6-s + (0.367 − 0.367i)7-s + 0.252·8-s + 0.145i·9-s + (−0.116 − 0.116i)11-s + (0.886 + 0.886i)12-s + 1.49i·13-s + (0.541 − 0.541i)14-s − 0.799·16-s + (−0.819 − 0.572i)17-s + 0.213i·18-s − 1.11i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 - 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.869 - 0.494i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.869 - 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.40001 + 0.899242i\)
\(L(\frac12)\) \(\approx\) \(3.40001 + 0.899242i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.38 + 2.36i)T \)
good2 \( 1 - 2.08T + 2T^{2} \)
3 \( 1 + (-1.31 - 1.31i)T + 3iT^{2} \)
7 \( 1 + (-0.971 + 0.971i)T - 7iT^{2} \)
11 \( 1 + (0.384 + 0.384i)T + 11iT^{2} \)
13 \( 1 - 5.39iT - 13T^{2} \)
19 \( 1 + 4.86iT - 19T^{2} \)
23 \( 1 + (-1.26 + 1.26i)T - 23iT^{2} \)
29 \( 1 + (1.29 - 1.29i)T - 29iT^{2} \)
31 \( 1 + (5.73 - 5.73i)T - 31iT^{2} \)
37 \( 1 + (-4.22 - 4.22i)T + 37iT^{2} \)
41 \( 1 + (2.70 + 2.70i)T + 41iT^{2} \)
43 \( 1 - 3.66T + 43T^{2} \)
47 \( 1 + 9.07iT - 47T^{2} \)
53 \( 1 - 10.5T + 53T^{2} \)
59 \( 1 - 6.52iT - 59T^{2} \)
61 \( 1 + (10.9 + 10.9i)T + 61iT^{2} \)
67 \( 1 - 5.68iT - 67T^{2} \)
71 \( 1 + (-0.749 + 0.749i)T - 71iT^{2} \)
73 \( 1 + (10.3 + 10.3i)T + 73iT^{2} \)
79 \( 1 + (-0.878 - 0.878i)T + 79iT^{2} \)
83 \( 1 - 13.5T + 83T^{2} \)
89 \( 1 - 0.989T + 89T^{2} \)
97 \( 1 + (-8.05 - 8.05i)T + 97iT^{2} \)
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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

−11.43298746781267548964061735943, −10.59836423784515693373421750544, −9.141813569127662224332021373323, −8.925853909970098537672238257347, −7.21127928498020536871309903646, −6.44917140543163931611320772518, −4.96962827011359248733870219314, −4.40306697893599544764131823076, −3.49519872253261796584709807914, −2.36571111800565000415472227211, 2.00676258610101639636567116562, 2.99112591469537480019549339482, 4.12446143213451351825118637143, 5.38489501076848187278286785769, 6.06627068763835988397042766777, 7.39469997429934234590145393688, 8.082833390055605139890840500259, 9.081108984291045723129052624262, 10.49803016406350202512746994576, 11.41123189110714874994576540397

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