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Properties

Label 2-425-17.13-c1-0-4
Degree $2$
Conductor $425$
Sign $-0.944 - 0.327i$
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  + 1.68i·2-s + (1.27 + 1.27i)3-s − 0.830·4-s + (−2.14 + 2.14i)6-s + (−1.92 + 1.92i)7-s + 1.96i·8-s + 0.249i·9-s + (−0.0173 + 0.0173i)11-s + (−1.05 − 1.05i)12-s + 3.62·13-s + (−3.24 − 3.24i)14-s − 4.97·16-s + (−3.90 + 1.33i)17-s − 0.419·18-s + 0.603i·19-s + ⋯
L(s)  = 1  + 1.18i·2-s + (0.735 + 0.735i)3-s − 0.415·4-s + (−0.875 + 0.875i)6-s + (−0.728 + 0.728i)7-s + 0.695i·8-s + 0.0830i·9-s + (−0.00524 + 0.00524i)11-s + (−0.305 − 0.305i)12-s + 1.00·13-s + (−0.866 − 0.866i)14-s − 1.24·16-s + (−0.946 + 0.323i)17-s − 0.0987·18-s + 0.138i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.944 - 0.327i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.944 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282613 + 1.67750i\)
\(L(\frac12)\) \(\approx\) \(0.282613 + 1.67750i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.90 - 1.33i)T \)
good2 \( 1 - 1.68iT - 2T^{2} \)
3 \( 1 + (-1.27 - 1.27i)T + 3iT^{2} \)
7 \( 1 + (1.92 - 1.92i)T - 7iT^{2} \)
11 \( 1 + (0.0173 - 0.0173i)T - 11iT^{2} \)
13 \( 1 - 3.62T + 13T^{2} \)
19 \( 1 - 0.603iT - 19T^{2} \)
23 \( 1 + (-1.94 + 1.94i)T - 23iT^{2} \)
29 \( 1 + (-3.01 - 3.01i)T + 29iT^{2} \)
31 \( 1 + (0.422 + 0.422i)T + 31iT^{2} \)
37 \( 1 + (-7.50 - 7.50i)T + 37iT^{2} \)
41 \( 1 + (-5.07 + 5.07i)T - 41iT^{2} \)
43 \( 1 + 12.0iT - 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 2.05iT - 53T^{2} \)
59 \( 1 + 0.926iT - 59T^{2} \)
61 \( 1 + (-8.41 + 8.41i)T - 61iT^{2} \)
67 \( 1 - 5.79T + 67T^{2} \)
71 \( 1 + (3.84 + 3.84i)T + 71iT^{2} \)
73 \( 1 + (-2.88 - 2.88i)T + 73iT^{2} \)
79 \( 1 + (-9.68 + 9.68i)T - 79iT^{2} \)
83 \( 1 - 12.1iT - 83T^{2} \)
89 \( 1 + 7.11T + 89T^{2} \)
97 \( 1 + (-5.01 - 5.01i)T + 97iT^{2} \)
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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

−11.48095734878408961810680718652, −10.50237611821564192309745886471, −9.346860442075471723268311248827, −8.763000572957161269038009777325, −8.115780675191957449781313634504, −6.68019588733635222225959363341, −6.19465019792869481268167325853, −4.97312596111349686732352829167, −3.72802802400603340031608880037, −2.53470155161430605108931351630, 1.07038090426878260619949525002, 2.40725948490643318158253827361, 3.32409029753353059268995043521, 4.39751252830110353611587318084, 6.31818665438502554196585366480, 7.07500064357585994204068076252, 8.100626074128502113965630360794, 9.178963238644997150485438093086, 9.942771538961050831882192607802, 10.99954456983189083452633412283

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