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Properties

Label 2-430-215.128-c1-0-11
Degree $2$
Conductor $430$
Sign $0.0452 - 0.998i$
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 + (−0.427 + 0.427i)3-s + 1.00i·4-s + (2.20 + 0.343i)5-s − 0.604·6-s + (−0.978 − 0.978i)7-s + (−0.707 + 0.707i)8-s + 2.63i·9-s + (1.31 + 1.80i)10-s − 0.412·11-s + (−0.427 − 0.427i)12-s + (3.54 + 3.54i)13-s − 1.38i·14-s + (−1.09 + 0.797i)15-s − 1.00·16-s + (1.54 − 1.54i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (−0.246 + 0.246i)3-s + 0.500i·4-s + (0.988 + 0.153i)5-s − 0.246·6-s + (−0.369 − 0.369i)7-s + (−0.250 + 0.250i)8-s + 0.878i·9-s + (0.417 + 0.570i)10-s − 0.124·11-s + (−0.123 − 0.123i)12-s + (0.982 + 0.982i)13-s − 0.369i·14-s + (−0.281 + 0.206i)15-s − 0.250·16-s + (0.374 − 0.374i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0452 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0452 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.0452 - 0.998i$
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: \((2,\ 430,\ (\ :1/2),\ 0.0452 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34283 + 1.28335i\)
\(L(\frac12)\) \(\approx\) \(1.34283 + 1.28335i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.20 - 0.343i)T \)
43 \( 1 + (-6.50 - 0.804i)T \)
good3 \( 1 + (0.427 - 0.427i)T - 3iT^{2} \)
7 \( 1 + (0.978 + 0.978i)T + 7iT^{2} \)
11 \( 1 + 0.412T + 11T^{2} \)
13 \( 1 + (-3.54 - 3.54i)T + 13iT^{2} \)
17 \( 1 + (-1.54 + 1.54i)T - 17iT^{2} \)
19 \( 1 + 2.28T + 19T^{2} \)
23 \( 1 + (-1.63 - 1.63i)T + 23iT^{2} \)
29 \( 1 + 1.44T + 29T^{2} \)
31 \( 1 + 2.27T + 31T^{2} \)
37 \( 1 + (2.91 + 2.91i)T + 37iT^{2} \)
41 \( 1 + 4.30T + 41T^{2} \)
47 \( 1 + (-1.12 + 1.12i)T - 47iT^{2} \)
53 \( 1 + (-4.87 - 4.87i)T + 53iT^{2} \)
59 \( 1 + 7.30iT - 59T^{2} \)
61 \( 1 + 4.90iT - 61T^{2} \)
67 \( 1 + (-2.15 + 2.15i)T - 67iT^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 + (-5.71 + 5.71i)T - 73iT^{2} \)
79 \( 1 + 10.4iT - 79T^{2} \)
83 \( 1 + (-0.526 - 0.526i)T + 83iT^{2} \)
89 \( 1 - 3.35T + 89T^{2} \)
97 \( 1 + (-2.30e-5 + 2.30e-5i)T - 97iT^{2} \)
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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

−11.21350371077717979316186870030, −10.59259599874474862146310938579, −9.566268691048380843409638022745, −8.692632413606022040147568498527, −7.44792090720771709373679133408, −6.54208573345766460288985846497, −5.69379763926597332992083290620, −4.77538895370002744943983674634, −3.55327038653855553491854100835, −2.01960307476934652327477388098, 1.17224240837253710278509710468, 2.69899723467275376248066073071, 3.83651683041514904116474652386, 5.41573764805183627910868636060, 5.97256902962704666037214055430, 6.82532678923677316338865576345, 8.452636201748339377185059283527, 9.270794564156282415174663971628, 10.20392876269337723662355260906, 10.89888383752608584497309288571

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