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Properties

Label 2-425-85.19-c1-0-24
Degree $2$
Conductor $425$
Sign $-0.848 - 0.528i$
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  + (1.09 − 1.09i)2-s + (−1.15 − 2.77i)3-s − 0.419i·4-s + (−4.32 − 1.79i)6-s + (−3.19 − 1.32i)7-s + (1.73 + 1.73i)8-s + (−4.27 + 4.27i)9-s + (−3.92 − 1.62i)11-s + (−1.16 + 0.483i)12-s + 0.127·13-s + (−4.96 + 2.05i)14-s + 4.66·16-s + (0.193 − 4.11i)17-s + 9.40i·18-s + (−1.81 − 1.81i)19-s + ⋯
L(s)  = 1  + (0.777 − 0.777i)2-s + (−0.664 − 1.60i)3-s − 0.209i·4-s + (−1.76 − 0.731i)6-s + (−1.20 − 0.499i)7-s + (0.614 + 0.614i)8-s + (−1.42 + 1.42i)9-s + (−1.18 − 0.489i)11-s + (−0.336 + 0.139i)12-s + 0.0353·13-s + (−1.32 + 0.549i)14-s + 1.16·16-s + (0.0468 − 0.998i)17-s + 2.21i·18-s + (−0.417 − 0.417i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.848 - 0.528i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ -0.848 - 0.528i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278843 + 0.975079i\)
\(L(\frac12)\) \(\approx\) \(0.278843 + 0.975079i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.193 + 4.11i)T \)
good2 \( 1 + (-1.09 + 1.09i)T - 2iT^{2} \)
3 \( 1 + (1.15 + 2.77i)T + (-2.12 + 2.12i)T^{2} \)
7 \( 1 + (3.19 + 1.32i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (3.92 + 1.62i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 - 0.127T + 13T^{2} \)
19 \( 1 + (1.81 + 1.81i)T + 19iT^{2} \)
23 \( 1 + (-1.24 + 3.00i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-1.87 - 4.53i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-4.95 + 2.05i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (0.677 + 1.63i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-3.85 + 9.29i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (1.79 + 1.79i)T + 43iT^{2} \)
47 \( 1 + 4.59T + 47T^{2} \)
53 \( 1 + (-1.15 + 1.15i)T - 53iT^{2} \)
59 \( 1 + (-4.34 + 4.34i)T - 59iT^{2} \)
61 \( 1 + (1.54 - 3.73i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + 6.88iT - 67T^{2} \)
71 \( 1 + (6.66 - 2.76i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (13.5 - 5.59i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.75 - 1.97i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \)
89 \( 1 - 0.600iT - 89T^{2} \)
97 \( 1 + (-7.09 + 2.93i)T + (68.5 - 68.5i)T^{2} \)
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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

−10.95936203196906266180767940893, −10.25763685894852466992426269810, −8.620641928037959646057885726302, −7.55926076430548854227821563484, −6.87966185322867216650353381198, −5.85959859904603681352865213882, −4.84174280849863631614505161165, −3.19258145530383497867413745915, −2.34993668814233669331597728526, −0.52374188253941021362318866029, 3.12040055019942836396081918676, 4.21257295970486329772306430847, 5.03188283846200327011698799871, 5.91678434856671129960860559937, 6.44216340262551403006813654420, 7.961484196767599509787967904374, 9.297012514194475460344567057999, 10.19621053812046514477109046871, 10.36089350255167513542645734464, 11.69085867855818363087342304604

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