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Properties

Label 2-430-215.128-c1-0-2
Degree $2$
Conductor $430$
Sign $-0.996 - 0.0800i$
Analytic cond. $3.43356$
Root an. cond. $1.85298$
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  + (0.707 + 0.707i)2-s + (−0.146 + 0.146i)3-s + 1.00i·4-s + (−1.92 + 1.14i)5-s − 0.206·6-s + (−2.59 − 2.59i)7-s + (−0.707 + 0.707i)8-s + 2.95i·9-s + (−2.16 − 0.548i)10-s − 1.90·11-s + (−0.146 − 0.146i)12-s + (2.03 + 2.03i)13-s − 3.67i·14-s + (0.113 − 0.448i)15-s − 1.00·16-s + (−5.27 + 5.27i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.0843 + 0.0843i)3-s + 0.500i·4-s + (−0.859 + 0.511i)5-s − 0.0843·6-s + (−0.980 − 0.980i)7-s + (−0.250 + 0.250i)8-s + 0.985i·9-s + (−0.685 − 0.173i)10-s − 0.575·11-s + (−0.0421 − 0.0421i)12-s + (0.564 + 0.564i)13-s − 0.980i·14-s + (0.0292 − 0.115i)15-s − 0.250·16-s + (−1.27 + 1.27i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0800i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.996 - 0.0800i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.996 - 0.0800i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0280230 + 0.698870i\)
\(L(\frac12)\) \(\approx\) \(0.0280230 + 0.698870i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (1.92 - 1.14i)T \)
43 \( 1 + (4.92 - 4.33i)T \)
good3 \( 1 + (0.146 - 0.146i)T - 3iT^{2} \)
7 \( 1 + (2.59 + 2.59i)T + 7iT^{2} \)
11 \( 1 + 1.90T + 11T^{2} \)
13 \( 1 + (-2.03 - 2.03i)T + 13iT^{2} \)
17 \( 1 + (5.27 - 5.27i)T - 17iT^{2} \)
19 \( 1 + 3.74T + 19T^{2} \)
23 \( 1 + (3.41 + 3.41i)T + 23iT^{2} \)
29 \( 1 - 0.492T + 29T^{2} \)
31 \( 1 - 9.89T + 31T^{2} \)
37 \( 1 + (-5.40 - 5.40i)T + 37iT^{2} \)
41 \( 1 - 4.38T + 41T^{2} \)
47 \( 1 + (0.662 - 0.662i)T - 47iT^{2} \)
53 \( 1 + (6.58 + 6.58i)T + 53iT^{2} \)
59 \( 1 - 9.59iT - 59T^{2} \)
61 \( 1 - 10.4iT - 61T^{2} \)
67 \( 1 + (-8.08 + 8.08i)T - 67iT^{2} \)
71 \( 1 - 16.3iT - 71T^{2} \)
73 \( 1 + (-7.69 + 7.69i)T - 73iT^{2} \)
79 \( 1 - 2.67iT - 79T^{2} \)
83 \( 1 + (1.34 + 1.34i)T + 83iT^{2} \)
89 \( 1 + 1.38T + 89T^{2} \)
97 \( 1 + (10.3 - 10.3i)T - 97iT^{2} \)
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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

−11.49003479229450428449121219570, −10.71378909560023613062607298685, −10.11108999659335414527473397324, −8.436789600088546809458917982997, −7.932353212063359586702711135623, −6.67341644455183880470117221840, −6.33137067313450798922188534600, −4.49598015942295122145131235675, −4.03016133331581533710252640724, −2.63058681233357555321707599201, 0.36220018659226168544063419600, 2.59964853644767109294009031436, 3.61162369115378660954724478345, 4.74256096337244937612492607806, 5.91801938561514404658819115751, 6.70569788855718676547998073676, 8.122330649376828387280446758333, 9.071634080769261059787063603929, 9.723289917112627763629294264821, 11.04466584760961303610091386448

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