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Properties

Label 2-430-215.108-c1-0-12
Degree $2$
Conductor $430$
Sign $0.0621 + 0.998i$
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  + (−0.330 − 0.943i)2-s + (−1.07 + 3.07i)3-s + (−0.781 + 0.623i)4-s + (−2.21 − 0.281i)5-s + 3.26·6-s + (−1.05 − 1.05i)7-s + (0.846 + 0.532i)8-s + (−5.96 − 4.75i)9-s + (0.467 + 2.18i)10-s + (−0.911 + 1.14i)11-s + (−1.07 − 3.07i)12-s + (3.00 − 4.78i)13-s + (−0.647 + 1.34i)14-s + (3.25 − 6.52i)15-s + (0.222 − 0.974i)16-s + (1.76 + 2.81i)17-s + ⋯
L(s)  = 1  + (−0.233 − 0.667i)2-s + (−0.621 + 1.77i)3-s + (−0.390 + 0.311i)4-s + (−0.992 − 0.125i)5-s + 1.33·6-s + (−0.399 − 0.399i)7-s + (0.299 + 0.188i)8-s + (−1.98 − 1.58i)9-s + (0.147 + 0.691i)10-s + (−0.274 + 0.344i)11-s + (−0.310 − 0.888i)12-s + (0.834 − 1.32i)13-s + (−0.173 + 0.359i)14-s + (0.840 − 1.68i)15-s + (0.0556 − 0.243i)16-s + (0.428 + 0.682i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0621 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0621 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.0621 + 0.998i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.0621 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.270917 - 0.254578i\)
\(L(\frac12)\) \(\approx\) \(0.270917 - 0.254578i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.330 + 0.943i)T \)
5 \( 1 + (2.21 + 0.281i)T \)
43 \( 1 + (5.51 + 3.54i)T \)
good3 \( 1 + (1.07 - 3.07i)T + (-2.34 - 1.87i)T^{2} \)
7 \( 1 + (1.05 + 1.05i)T + 7iT^{2} \)
11 \( 1 + (0.911 - 1.14i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (-3.00 + 4.78i)T + (-5.64 - 11.7i)T^{2} \)
17 \( 1 + (-1.76 - 2.81i)T + (-7.37 + 15.3i)T^{2} \)
19 \( 1 + (-0.343 - 0.430i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (-7.74 + 0.872i)T + (22.4 - 5.11i)T^{2} \)
29 \( 1 + (8.14 + 3.92i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (5.03 + 2.42i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (5.51 + 5.51i)T + 37iT^{2} \)
41 \( 1 + (4.34 + 2.09i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (-9.40 - 1.05i)T + (45.8 + 10.4i)T^{2} \)
53 \( 1 + (2.99 - 1.87i)T + (22.9 - 47.7i)T^{2} \)
59 \( 1 + (1.17 + 0.267i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (4.48 + 9.32i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (6.58 + 0.742i)T + (65.3 + 14.9i)T^{2} \)
71 \( 1 + (-0.437 + 0.348i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (1.29 + 0.812i)T + (31.6 + 65.7i)T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 + (3.30 + 1.15i)T + (64.8 + 51.7i)T^{2} \)
89 \( 1 + (0.815 - 0.392i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-0.294 - 2.61i)T + (-94.5 + 21.5i)T^{2} \)
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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

−10.68330356856817220674508563337, −10.45448436161623678357237891813, −9.329441592662549637696800428920, −8.598861462498531979071151421338, −7.45023633183109289061057120011, −5.76926197125119522802294292032, −4.89943002399522097396303365452, −3.69772675003890184454909822802, −3.41379009942211420893639394064, −0.30407969104062785249890654446, 1.35248702344837642045740911574, 3.18345215195130631483316471155, 5.04162054131087158540028120016, 6.01719278344421462348533579249, 7.07995430243377766750098543588, 7.22129834786631457347922436372, 8.457405329941377874220932535434, 9.060881494817000400034647722755, 10.92534124598052358421955812863, 11.45118180925847788140402689997

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