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Properties

Label 2-430-215.109-c1-0-20
Degree $2$
Conductor $430$
Sign $0.266 + 0.963i$
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  + (0.781 − 0.623i)2-s + (3.05 − 1.19i)3-s + (0.222 − 0.974i)4-s + (−2.16 + 0.546i)5-s + (1.64 − 2.84i)6-s + (0.798 − 0.460i)7-s + (−0.433 − 0.900i)8-s + (5.70 − 5.29i)9-s + (−1.35 + 1.77i)10-s + (−0.268 − 1.17i)11-s + (−0.489 − 3.24i)12-s + (−3.17 + 4.65i)13-s + (0.336 − 0.858i)14-s + (−5.97 + 4.27i)15-s + (−0.900 − 0.433i)16-s + (−1.38 + 0.103i)17-s + ⋯
L(s)  = 1  + (0.552 − 0.440i)2-s + (1.76 − 0.692i)3-s + (0.111 − 0.487i)4-s + (−0.969 + 0.244i)5-s + (0.670 − 1.16i)6-s + (0.301 − 0.174i)7-s + (−0.153 − 0.318i)8-s + (1.90 − 1.76i)9-s + (−0.428 + 0.562i)10-s + (−0.0810 − 0.355i)11-s + (−0.141 − 0.937i)12-s + (−0.881 + 1.29i)13-s + (0.0900 − 0.229i)14-s + (−1.54 + 1.10i)15-s + (−0.225 − 0.108i)16-s + (−0.335 + 0.0251i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.266 + 0.963i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.266 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21366 - 1.68432i\)
\(L(\frac12)\) \(\approx\) \(2.21366 - 1.68432i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 + 0.623i)T \)
5 \( 1 + (2.16 - 0.546i)T \)
43 \( 1 + (-2.06 - 6.22i)T \)
good3 \( 1 + (-3.05 + 1.19i)T + (2.19 - 2.04i)T^{2} \)
7 \( 1 + (-0.798 + 0.460i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.268 + 1.17i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (3.17 - 4.65i)T + (-4.74 - 12.1i)T^{2} \)
17 \( 1 + (1.38 - 0.103i)T + (16.8 - 2.53i)T^{2} \)
19 \( 1 + (-2.66 - 2.47i)T + (1.41 + 18.9i)T^{2} \)
23 \( 1 + (-0.0390 + 0.126i)T + (-19.0 - 12.9i)T^{2} \)
29 \( 1 + (3.26 - 8.30i)T + (-21.2 - 19.7i)T^{2} \)
31 \( 1 + (-5.63 + 0.849i)T + (29.6 - 9.13i)T^{2} \)
37 \( 1 + (1.64 + 0.952i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.73 + 3.43i)T + (-9.12 + 39.9i)T^{2} \)
47 \( 1 + (5.20 + 1.18i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (5.27 + 7.73i)T + (-19.3 + 49.3i)T^{2} \)
59 \( 1 + (-11.6 - 5.59i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (9.57 + 1.44i)T + (58.2 + 17.9i)T^{2} \)
67 \( 1 + (0.758 - 0.817i)T + (-5.00 - 66.8i)T^{2} \)
71 \( 1 + (-13.2 + 4.09i)T + (58.6 - 39.9i)T^{2} \)
73 \( 1 + (3.72 - 5.47i)T + (-26.6 - 67.9i)T^{2} \)
79 \( 1 + (5.12 + 8.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.58 - 0.620i)T + (60.8 - 56.4i)T^{2} \)
89 \( 1 + (1.69 + 4.31i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (2.10 - 0.479i)T + (87.3 - 42.0i)T^{2} \)
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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

−11.21887793893764199980880878651, −9.938231855262900047543216125928, −9.051449686070665454083278406343, −8.189039931262617424857527924679, −7.33898225772564858479517225917, −6.67280460900416395387958871105, −4.66682580047118159227563256834, −3.72019474517767424350975585633, −2.85987466697750971718323987885, −1.62925999436326009179155632571, 2.50438775899769220817522709328, 3.39465387588883419525074324537, 4.43437157745556299524029855945, 5.12772914896123488256291694984, 7.11459389892035254694953403246, 7.909946741494706584520055945691, 8.299766160891342499668094816303, 9.382352266973792165378698206612, 10.20111460134360708823385176423, 11.42028784707832731136365770039

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