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Properties

Label 2-425-17.16-c1-0-12
Degree $2$
Conductor $425$
Sign $0.395 - 0.918i$
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  + 2.17·2-s + 2.21i·3-s + 2.70·4-s + 4.80i·6-s − 1.02i·7-s + 1.53·8-s − 1.90·9-s + 4.98i·11-s + 6.00i·12-s + 3.87·13-s − 2.21i·14-s − 2.07·16-s + (1.63 − 3.78i)17-s − 4.14·18-s − 4.04·19-s + ⋯
L(s)  = 1  + 1.53·2-s + 1.27i·3-s + 1.35·4-s + 1.96i·6-s − 0.385i·7-s + 0.544·8-s − 0.636·9-s + 1.50i·11-s + 1.73i·12-s + 1.07·13-s − 0.592i·14-s − 0.519·16-s + (0.395 − 0.918i)17-s − 0.976·18-s − 0.929·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.395 - 0.918i$
Analytic conductor: \(3.39364\)
Root analytic conductor: \(1.84218\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{425} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 425,\ (\ :1/2),\ 0.395 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.53894 + 1.67093i\)
\(L(\frac12)\) \(\approx\) \(2.53894 + 1.67093i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 + 3.78i)T \)
good2 \( 1 - 2.17T + 2T^{2} \)
3 \( 1 - 2.21iT - 3T^{2} \)
7 \( 1 + 1.02iT - 7T^{2} \)
11 \( 1 - 4.98iT - 11T^{2} \)
13 \( 1 - 3.87T + 13T^{2} \)
19 \( 1 + 4.04T + 19T^{2} \)
23 \( 1 + 7.02iT - 23T^{2} \)
29 \( 1 + 0.644iT - 29T^{2} \)
31 \( 1 + 4.25iT - 31T^{2} \)
37 \( 1 + 10.2iT - 37T^{2} \)
41 \( 1 - 5.82iT - 41T^{2} \)
43 \( 1 - 5.86T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 11.3T + 53T^{2} \)
59 \( 1 + 0.447T + 59T^{2} \)
61 \( 1 + 3.14iT - 61T^{2} \)
67 \( 1 + 5.07T + 67T^{2} \)
71 \( 1 - 3.41iT - 71T^{2} \)
73 \( 1 - 3.78iT - 73T^{2} \)
79 \( 1 + 0.376iT - 79T^{2} \)
83 \( 1 + 3.57T + 83T^{2} \)
89 \( 1 + 2.63T + 89T^{2} \)
97 \( 1 - 10.9iT - 97T^{2} \)
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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

−11.36151599492259332733032359253, −10.61446524389363358381586866416, −9.774572919559425188196852351784, −8.822388100771552330145172185292, −7.31090971259353292910481397607, −6.30685877151795261009865281654, −5.18749484390966921189980436230, −4.32987479106984200799021775873, −3.91462519577942106421140241392, −2.49190393518664640539965398266, 1.53256146755489070250067021793, 3.00859938335842069766558251773, 3.94007947996872996627297978573, 5.55746968985652384823615754350, 6.06805043545206972453595156147, 6.84390375350778083774579211767, 8.127626760751069265928581179162, 8.820310416304616188003109325936, 10.60802026198560797544733453085, 11.49617175947372065783619207556

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