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Properties

Label 2-425-17.13-c1-0-7
Degree $2$
Conductor $425$
Sign $0.836 - 0.547i$
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  − 0.677i·2-s + (1.66 + 1.66i)3-s + 1.54·4-s + (1.12 − 1.12i)6-s + (−3.02 + 3.02i)7-s − 2.39i·8-s + 2.55i·9-s + (−1.17 + 1.17i)11-s + (2.56 + 2.56i)12-s + 6.21·13-s + (2.04 + 2.04i)14-s + 1.45·16-s + (1.32 − 3.90i)17-s + 1.73·18-s + 3.38i·19-s + ⋯
L(s)  = 1  − 0.479i·2-s + (0.962 + 0.962i)3-s + 0.770·4-s + (0.461 − 0.461i)6-s + (−1.14 + 1.14i)7-s − 0.848i·8-s + 0.852i·9-s + (−0.353 + 0.353i)11-s + (0.741 + 0.741i)12-s + 1.72·13-s + (0.547 + 0.547i)14-s + 0.363·16-s + (0.322 − 0.946i)17-s + 0.408·18-s + 0.777i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(425\)    =    \(5^{2} \cdot 17\)
Sign: $0.836 - 0.547i$
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: \((2,\ 425,\ (\ :1/2),\ 0.836 - 0.547i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.99298 + 0.594198i\)
\(L(\frac12)\) \(\approx\) \(1.99298 + 0.594198i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 3.90i)T \)
good2 \( 1 + 0.677iT - 2T^{2} \)
3 \( 1 + (-1.66 - 1.66i)T + 3iT^{2} \)
7 \( 1 + (3.02 - 3.02i)T - 7iT^{2} \)
11 \( 1 + (1.17 - 1.17i)T - 11iT^{2} \)
13 \( 1 - 6.21T + 13T^{2} \)
19 \( 1 - 3.38iT - 19T^{2} \)
23 \( 1 + (3.30 - 3.30i)T - 23iT^{2} \)
29 \( 1 + (2.57 + 2.57i)T + 29iT^{2} \)
31 \( 1 + (2.12 + 2.12i)T + 31iT^{2} \)
37 \( 1 + (3.78 + 3.78i)T + 37iT^{2} \)
41 \( 1 + (1.54 - 1.54i)T - 41iT^{2} \)
43 \( 1 - 0.998iT - 43T^{2} \)
47 \( 1 - 2.00T + 47T^{2} \)
53 \( 1 + 6.95iT - 53T^{2} \)
59 \( 1 + 6.30iT - 59T^{2} \)
61 \( 1 + (-4.62 + 4.62i)T - 61iT^{2} \)
67 \( 1 + 5.80T + 67T^{2} \)
71 \( 1 + (9.60 + 9.60i)T + 71iT^{2} \)
73 \( 1 + (-7.01 - 7.01i)T + 73iT^{2} \)
79 \( 1 + (-0.820 + 0.820i)T - 79iT^{2} \)
83 \( 1 - 3.65iT - 83T^{2} \)
89 \( 1 - 2.69T + 89T^{2} \)
97 \( 1 + (7.40 + 7.40i)T + 97iT^{2} \)
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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

−11.19977910507634807707691988002, −10.12393850474769000771093482139, −9.607564196444263651385233531183, −8.833850946241181927870113605401, −7.79414904120272081171645765357, −6.43119315687844955121445557995, −5.59795490248876858082095945841, −3.75600823870450447334562151096, −3.24758644841945179823625589409, −2.13474387297100110461170401022, 1.40601133373346756352026358988, 2.90247002571030957063690467903, 3.77545822530081132879483336126, 5.87545401671839559063618255862, 6.64845431758303049384458326769, 7.28472703745901452976369932934, 8.200388617764208340277865296220, 8.863207665217266659802629551292, 10.41069849397256271634572782113, 10.86120646173477960253506116256

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