Properties

Label 2-430-43.10-c1-0-15
Degree $2$
Conductor $430$
Sign $-0.477 - 0.878i$
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  + (−0.222 − 0.974i)2-s + (−0.861 − 0.799i)3-s + (−0.900 + 0.433i)4-s + (−0.988 + 0.149i)5-s + (−0.587 + 1.01i)6-s + (−1.53 − 2.65i)7-s + (0.623 + 0.781i)8-s + (−0.121 − 1.61i)9-s + (0.365 + 0.930i)10-s + (1.59 + 0.770i)11-s + (1.12 + 0.346i)12-s + (−0.784 + 1.99i)13-s + (−2.24 + 2.08i)14-s + (0.970 + 0.661i)15-s + (0.623 − 0.781i)16-s + (−5.40 − 0.814i)17-s + ⋯
L(s)  = 1  + (−0.157 − 0.689i)2-s + (−0.497 − 0.461i)3-s + (−0.450 + 0.216i)4-s + (−0.442 + 0.0666i)5-s + (−0.239 + 0.415i)6-s + (−0.579 − 1.00i)7-s + (0.220 + 0.276i)8-s + (−0.0403 − 0.538i)9-s + (0.115 + 0.294i)10-s + (0.482 + 0.232i)11-s + (0.324 + 0.0999i)12-s + (−0.217 + 0.554i)13-s + (−0.600 + 0.557i)14-s + (0.250 + 0.170i)15-s + (0.155 − 0.195i)16-s + (−1.31 − 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.477 - 0.878i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0802332 + 0.134933i\)
\(L(\frac12)\) \(\approx\) \(0.0802332 + 0.134933i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 + 0.974i)T \)
5 \( 1 + (0.988 - 0.149i)T \)
43 \( 1 + (5.48 + 3.59i)T \)
good3 \( 1 + (0.861 + 0.799i)T + (0.224 + 2.99i)T^{2} \)
7 \( 1 + (1.53 + 2.65i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.59 - 0.770i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (0.784 - 1.99i)T + (-9.52 - 8.84i)T^{2} \)
17 \( 1 + (5.40 + 0.814i)T + (16.2 + 5.01i)T^{2} \)
19 \( 1 + (0.568 - 7.58i)T + (-18.7 - 2.83i)T^{2} \)
23 \( 1 + (5.24 - 3.57i)T + (8.40 - 21.4i)T^{2} \)
29 \( 1 + (-2.13 + 1.98i)T + (2.16 - 28.9i)T^{2} \)
31 \( 1 + (4.97 + 1.53i)T + (25.6 + 17.4i)T^{2} \)
37 \( 1 + (-2.20 + 3.82i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.306 + 1.34i)T + (-36.9 + 17.7i)T^{2} \)
47 \( 1 + (-4.13 + 1.99i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-0.102 - 0.261i)T + (-38.8 + 36.0i)T^{2} \)
59 \( 1 + (9.41 - 11.8i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-11.0 + 3.39i)T + (50.4 - 34.3i)T^{2} \)
67 \( 1 + (-1.13 + 15.1i)T + (-66.2 - 9.98i)T^{2} \)
71 \( 1 + (8.01 + 5.46i)T + (25.9 + 66.0i)T^{2} \)
73 \( 1 + (1.33 - 3.40i)T + (-53.5 - 49.6i)T^{2} \)
79 \( 1 + (-0.640 - 1.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (11.1 + 10.3i)T + (6.20 + 82.7i)T^{2} \)
89 \( 1 + (6.63 + 6.15i)T + (6.65 + 88.7i)T^{2} \)
97 \( 1 + (-7.66 - 3.69i)T + (60.4 + 75.8i)T^{2} \)
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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

−10.62264930764554670515286821683, −9.807197817702008328634143873675, −8.943095947451089431892407342028, −7.66015748110520625175105172133, −6.87069213492935454288021110909, −5.93682152940757673072500452148, −4.21033563861921781281611188692, −3.64906177986880942548613824393, −1.75677616588901985366632554412, −0.10880244355402091780153744823, 2.62201078857205642279555254287, 4.27982340204594822627949600821, 5.13787693413425200690425154121, 6.14620217239104341594104848247, 6.96583032554359878407587028323, 8.285126110621417414930475285562, 8.900174343582971932034473820987, 9.854321650884443900691651688235, 10.89170234242202175112204223863, 11.59027041265791210657495066191

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