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Properties

Label 2-430-215.108-c1-0-15
Degree $2$
Conductor $430$
Sign $0.256 + 0.966i$
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 + (0.210 − 0.601i)3-s + (−0.781 + 0.623i)4-s + (1.96 − 1.06i)5-s − 0.636·6-s + (0.415 + 0.415i)7-s + (0.846 + 0.532i)8-s + (2.02 + 1.61i)9-s + (−1.65 − 1.50i)10-s + (2.96 − 3.71i)11-s + (0.210 + 0.601i)12-s + (−1.94 + 3.09i)13-s + (0.254 − 0.528i)14-s + (−0.228 − 1.40i)15-s + (0.222 − 0.974i)16-s + (1.63 + 2.60i)17-s + ⋯
L(s)  = 1  + (−0.233 − 0.667i)2-s + (0.121 − 0.347i)3-s + (−0.390 + 0.311i)4-s + (0.878 − 0.477i)5-s − 0.259·6-s + (0.156 + 0.156i)7-s + (0.299 + 0.188i)8-s + (0.676 + 0.539i)9-s + (−0.523 − 0.475i)10-s + (0.894 − 1.12i)11-s + (0.0607 + 0.173i)12-s + (−0.539 + 0.858i)13-s + (0.0680 − 0.141i)14-s + (−0.0588 − 0.362i)15-s + (0.0556 − 0.243i)16-s + (0.397 + 0.632i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $0.256 + 0.966i$
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: \((2,\ 430,\ (\ :1/2),\ 0.256 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22129 - 0.939829i\)
\(L(\frac12)\) \(\approx\) \(1.22129 - 0.939829i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.96 + 1.06i)T \)
43 \( 1 + (-3.55 - 5.51i)T \)
good3 \( 1 + (-0.210 + 0.601i)T + (-2.34 - 1.87i)T^{2} \)
7 \( 1 + (-0.415 - 0.415i)T + 7iT^{2} \)
11 \( 1 + (-2.96 + 3.71i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.94 - 3.09i)T + (-5.64 - 11.7i)T^{2} \)
17 \( 1 + (-1.63 - 2.60i)T + (-7.37 + 15.3i)T^{2} \)
19 \( 1 + (2.26 + 2.83i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (-4.65 + 0.524i)T + (22.4 - 5.11i)T^{2} \)
29 \( 1 + (4.39 + 2.11i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 + (7.63 + 3.67i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (-2.70 - 2.70i)T + 37iT^{2} \)
41 \( 1 + (9.47 + 4.56i)T + (25.5 + 32.0i)T^{2} \)
47 \( 1 + (-3.54 - 0.399i)T + (45.8 + 10.4i)T^{2} \)
53 \( 1 + (10.3 - 6.50i)T + (22.9 - 47.7i)T^{2} \)
59 \( 1 + (-13.6 - 3.11i)T + (53.1 + 25.5i)T^{2} \)
61 \( 1 + (-2.09 - 4.35i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (-3.81 - 0.430i)T + (65.3 + 14.9i)T^{2} \)
71 \( 1 + (3.80 - 3.03i)T + (15.7 - 69.2i)T^{2} \)
73 \( 1 + (13.5 + 8.48i)T + (31.6 + 65.7i)T^{2} \)
79 \( 1 - 9.00iT - 79T^{2} \)
83 \( 1 + (5.45 + 1.90i)T + (64.8 + 51.7i)T^{2} \)
89 \( 1 + (1.11 - 0.534i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-1.42 - 12.6i)T + (-94.5 + 21.5i)T^{2} \)
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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

−11.00894319421864727413250581547, −10.03907074342077060794231701857, −9.131544156941264748801036829942, −8.598337835105427301651086171332, −7.33573368753143167240563064358, −6.25445092741966682451574047033, −5.08338687388585409883685125983, −3.95216733255257731933573858995, −2.34011242676651755704989287889, −1.32207984017815720646462533406, 1.64363321203977636950032181293, 3.43481044253586377431272479261, 4.73186152849867854433019558006, 5.69812917626588437953777153781, 6.95851597282232750538780119438, 7.28333230288647496139747682518, 8.806878045871816828863578795296, 9.678661342003935337628671595254, 10.02974845948082951317576135033, 11.07150158514992021841042918168

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