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Properties

Label 2-425-5.4-c3-0-68
Degree $2$
Conductor $425$
Sign $-0.894 - 0.447i$
Analytic cond. $25.0758$
Root an. cond. $5.00757$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·2-s − 10i·3-s − 4-s + 30·6-s − 22i·7-s + 21i·8-s − 73·9-s − 30·11-s + 10i·12-s + 46i·13-s + 66·14-s − 71·16-s + 17i·17-s − 219i·18-s − 104·19-s + ⋯
L(s)  = 1  + 1.06i·2-s − 1.92i·3-s − 0.125·4-s + 2.04·6-s − 1.18i·7-s + 0.928i·8-s − 2.70·9-s − 0.822·11-s + 0.240i·12-s + 0.981i·13-s + 1.25·14-s − 1.10·16-s + 0.242i·17-s − 2.86i·18-s − 1.25·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(25.0758\)
Root analytic conductor: \(5.00757\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 425,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - 17iT \)
good2 \( 1 - 3iT - 8T^{2} \)
3 \( 1 + 10iT - 27T^{2} \)
7 \( 1 + 22iT - 343T^{2} \)
11 \( 1 + 30T + 1.33e3T^{2} \)
13 \( 1 - 46iT - 2.19e3T^{2} \)
19 \( 1 + 104T + 6.85e3T^{2} \)
23 \( 1 + 42iT - 1.21e4T^{2} \)
29 \( 1 - 66T + 2.43e4T^{2} \)
31 \( 1 - 194T + 2.97e4T^{2} \)
37 \( 1 - 206iT - 5.06e4T^{2} \)
41 \( 1 + 126T + 6.89e4T^{2} \)
43 \( 1 - 388iT - 7.95e4T^{2} \)
47 \( 1 + 540iT - 1.03e5T^{2} \)
53 \( 1 + 78iT - 1.48e5T^{2} \)
59 \( 1 + 432T + 2.05e5T^{2} \)
61 \( 1 + 610T + 2.26e5T^{2} \)
67 \( 1 - 848iT - 3.00e5T^{2} \)
71 \( 1 + 174T + 3.57e5T^{2} \)
73 \( 1 + 362iT - 3.89e5T^{2} \)
79 \( 1 + 398T + 4.93e5T^{2} \)
83 \( 1 + 828iT - 5.71e5T^{2} \)
89 \( 1 + 630T + 7.04e5T^{2} \)
97 \( 1 + 1.48e3iT - 9.12e5T^{2} \)
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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

−10.41243398525853930244559124135, −8.605255457985561724442613959472, −8.084760451494656375176077203794, −7.24903915941787346754182018648, −6.66260908414996070240397985392, −6.06553492985283713231707035219, −4.66357555434667859352350636077, −2.67804981185690727043470448689, −1.53445848275965997101231738413, 0, 2.49262090035755248090166009305, 3.08168957013147942596082013563, 4.26135670094577363948482707570, 5.25095487018032995576094575912, 6.12300832119409654756342597977, 8.096343792674737553908227858991, 9.002504410272909045244761497841, 9.717582541826570954763122946192, 10.61406195755511322340145654864

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