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Properties

Label 2-430-215.79-c1-0-10
Degree $2$
Conductor $430$
Sign $0.975 + 0.222i$
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 + (−1.92 + 1.11i)3-s − 4-s + (−2.17 − 0.528i)5-s + (−1.11 − 1.92i)6-s + (−1.64 − 0.950i)7-s − i·8-s + (0.966 − 1.67i)9-s + (0.528 − 2.17i)10-s + 2.76·11-s + (1.92 − 1.11i)12-s + (1.72 + 0.996i)13-s + (0.950 − 1.64i)14-s + (4.76 − 1.39i)15-s + 16-s + (−2.11 − 1.22i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−1.11 + 0.641i)3-s − 0.5·4-s + (−0.971 − 0.236i)5-s + (−0.453 − 0.785i)6-s + (−0.622 − 0.359i)7-s − 0.353i·8-s + (0.322 − 0.557i)9-s + (0.167 − 0.687i)10-s + 0.834·11-s + (0.555 − 0.320i)12-s + (0.478 + 0.276i)13-s + (0.254 − 0.440i)14-s + (1.23 − 0.360i)15-s + 0.250·16-s + (−0.512 − 0.295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.222i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.222i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.975 + 0.222i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ 0.975 + 0.222i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.493589 - 0.0555148i\)
\(L(\frac12)\) \(\approx\) \(0.493589 - 0.0555148i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.17 + 0.528i)T \)
43 \( 1 + (2.06 + 6.22i)T \)
good3 \( 1 + (1.92 - 1.11i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.64 + 0.950i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 2.76T + 11T^{2} \)
13 \( 1 + (-1.72 - 0.996i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.11 + 1.22i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.26 + 2.18i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-7.86 + 4.54i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.47 - 2.56i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.748 + 1.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.00 - 2.88i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 2.10T + 41T^{2} \)
47 \( 1 + 8.63iT - 47T^{2} \)
53 \( 1 + (-7.16 + 4.13i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 0.157T + 59T^{2} \)
61 \( 1 + (-2.76 + 4.79i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.59 - 2.65i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.80 + 13.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (3.08 + 1.78i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.13 - 1.96i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.45 - 1.99i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.70 + 6.42i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.95iT - 97T^{2} \)
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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

−11.13149857050292273920030472825, −10.37467082286082189117124744417, −9.162727528571028495308132729380, −8.517694840339037442375308333015, −6.97831099792987839646664594750, −6.63933224182491626185935126722, −5.27989292789697766736015489049, −4.48140959556644269714962144733, −3.57346718625463359422907133638, −0.44204402671649269457997069924, 1.17739218827654720060314255528, 3.11024285313108978888367195791, 4.17024464473163847982135613464, 5.54456713687474029421915628718, 6.47983332264486582438501563345, 7.31441912594735950267420716361, 8.575597180771080705826065388240, 9.455180009782893089025175372612, 10.81505420208462894351009505322, 11.23081931490211429729699687000

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