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Properties

Label 2-430-215.128-c1-0-17
Degree $2$
Conductor $430$
Sign $0.999 + 0.0417i$
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 + (2.27 − 2.27i)3-s + 1.00i·4-s + (1.31 + 1.80i)5-s + 3.21·6-s + (0.136 + 0.136i)7-s + (−0.707 + 0.707i)8-s − 7.31i·9-s + (−0.346 + 2.20i)10-s − 0.331·11-s + (2.27 + 2.27i)12-s + (−0.227 − 0.227i)13-s + 0.192i·14-s + (7.09 + 1.11i)15-s − 1.00·16-s + (−2.17 + 2.17i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (1.31 − 1.31i)3-s + 0.500i·4-s + (0.588 + 0.808i)5-s + 1.31·6-s + (0.0514 + 0.0514i)7-s + (−0.250 + 0.250i)8-s − 2.43i·9-s + (−0.109 + 0.698i)10-s − 0.100·11-s + (0.655 + 0.655i)12-s + (−0.0631 − 0.0631i)13-s + 0.0514i·14-s + (1.83 + 0.287i)15-s − 0.250·16-s + (−0.528 + 0.528i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.999 + 0.0417i$
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: \((2,\ 430,\ (\ :1/2),\ 0.999 + 0.0417i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.76090 - 0.0576931i\)
\(L(\frac12)\) \(\approx\) \(2.76090 - 0.0576931i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.31 - 1.80i)T \)
43 \( 1 + (-4.05 - 5.15i)T \)
good3 \( 1 + (-2.27 + 2.27i)T - 3iT^{2} \)
7 \( 1 + (-0.136 - 0.136i)T + 7iT^{2} \)
11 \( 1 + 0.331T + 11T^{2} \)
13 \( 1 + (0.227 + 0.227i)T + 13iT^{2} \)
17 \( 1 + (2.17 - 2.17i)T - 17iT^{2} \)
19 \( 1 + 6.24T + 19T^{2} \)
23 \( 1 + (-1.18 - 1.18i)T + 23iT^{2} \)
29 \( 1 + 0.0611T + 29T^{2} \)
31 \( 1 - 5.84T + 31T^{2} \)
37 \( 1 + (6.41 + 6.41i)T + 37iT^{2} \)
41 \( 1 - 3.48T + 41T^{2} \)
47 \( 1 + (-5.80 + 5.80i)T - 47iT^{2} \)
53 \( 1 + (9.01 + 9.01i)T + 53iT^{2} \)
59 \( 1 - 5.56iT - 59T^{2} \)
61 \( 1 + 1.13iT - 61T^{2} \)
67 \( 1 + (8.76 - 8.76i)T - 67iT^{2} \)
71 \( 1 - 12.3iT - 71T^{2} \)
73 \( 1 + (8.28 - 8.28i)T - 73iT^{2} \)
79 \( 1 - 4.93iT - 79T^{2} \)
83 \( 1 + (-7.28 - 7.28i)T + 83iT^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 + (-10.1 + 10.1i)T - 97iT^{2} \)
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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

−11.33943916149665288867059314685, −10.13105527553617104717979875336, −8.920307177255883692206902744253, −8.306948473523076664661834076309, −7.26627596863848428513586927956, −6.68475975123194700952603403147, −5.84946185109768766152351518456, −3.99945678771452911635804502588, −2.81602409283307049220189896656, −1.96945376978655680237487783791, 2.05681770487920548839230707252, 3.06844450858772855739597594226, 4.47360555685213828779570825200, 4.71178136652478409757955949339, 6.14705935408993875174298263152, 7.86357072529150407162179811642, 8.928946633094142962976181228311, 9.220675014862585037299442634115, 10.32298690078935531902416175044, 10.76200634668743705820891923439

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