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Properties

Label 2-430-215.49-c1-0-15
Degree $2$
Conductor $430$
Sign $0.962 + 0.273i$
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  + i·2-s + (0.377 + 0.217i)3-s − 4-s + (−0.497 − 2.18i)5-s + (−0.217 + 0.377i)6-s + (−0.236 + 0.136i)7-s − i·8-s + (−1.40 − 2.43i)9-s + (2.18 − 0.497i)10-s + 3.88·11-s + (−0.377 − 0.217i)12-s + (0.559 − 0.323i)13-s + (−0.136 − 0.236i)14-s + (0.287 − 0.931i)15-s + 16-s + (1.73 − 1.00i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.217 + 0.125i)3-s − 0.5·4-s + (−0.222 − 0.974i)5-s + (−0.0889 + 0.154i)6-s + (−0.0893 + 0.0515i)7-s − 0.353i·8-s + (−0.468 − 0.811i)9-s + (0.689 − 0.157i)10-s + 1.17·11-s + (−0.108 − 0.0629i)12-s + (0.155 − 0.0895i)13-s + (−0.0364 − 0.0631i)14-s + (0.0742 − 0.240i)15-s + 0.250·16-s + (0.421 − 0.243i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.962 + 0.273i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.962 + 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31991 - 0.183677i\)
\(L(\frac12)\) \(\approx\) \(1.31991 - 0.183677i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (0.497 + 2.18i)T \)
43 \( 1 + (4.54 + 4.73i)T \)
good3 \( 1 + (-0.377 - 0.217i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.236 - 0.136i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 3.88T + 11T^{2} \)
13 \( 1 + (-0.559 + 0.323i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.73 + 1.00i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.66 + 6.35i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.31 - 1.33i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.52 + 2.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.240 - 0.417i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.19 - 2.42i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.71T + 41T^{2} \)
47 \( 1 - 4.57iT - 47T^{2} \)
53 \( 1 + (0.0419 + 0.0241i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 4.81T + 59T^{2} \)
61 \( 1 + (-3.10 - 5.37i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.80 + 5.08i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.77 - 9.99i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.41 + 3.12i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.35 + 2.35i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (9.28 + 5.36i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.0807 - 0.139i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.1iT - 97T^{2} \)
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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

−11.42499620615443084376589920870, −9.697637853451250364636409805851, −9.181219294258919898184056198775, −8.540516796236824787167435729938, −7.42309401082332581747245705470, −6.42804642023929039544062637563, −5.40814864352621093041210617149, −4.37001854507416742333661527165, −3.27692438906646331666394474662, −0.929148707445000960981402106744, 1.72136567373878876306073756979, 3.10603592168764116636495214823, 3.90807210022349264879936723123, 5.41892794588640856681989882305, 6.55153113479227228591685888247, 7.64425392448965079041413577400, 8.481168142812906287256220853134, 9.627910864836302186572541316805, 10.37208479154016888112704624165, 11.29102203676703512576846906049

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