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Properties

Label 2-430-215.109-c1-0-17
Degree $2$
Conductor $430$
Sign $-0.231 + 0.972i$
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  + (0.781 − 0.623i)2-s + (0.236 − 0.0929i)3-s + (0.222 − 0.974i)4-s + (−1.99 − 1.00i)5-s + (0.127 − 0.220i)6-s + (2.97 − 1.71i)7-s + (−0.433 − 0.900i)8-s + (−2.15 + 1.99i)9-s + (−2.18 + 0.456i)10-s + (−0.468 − 2.05i)11-s + (−0.0379 − 0.251i)12-s + (3.75 − 5.50i)13-s + (1.25 − 3.19i)14-s + (−0.566 − 0.0532i)15-s + (−0.900 − 0.433i)16-s + (−6.38 + 0.478i)17-s + ⋯
L(s)  = 1  + (0.552 − 0.440i)2-s + (0.136 − 0.0536i)3-s + (0.111 − 0.487i)4-s + (−0.892 − 0.450i)5-s + (0.0519 − 0.0899i)6-s + (1.12 − 0.648i)7-s + (−0.153 − 0.318i)8-s + (−0.717 + 0.665i)9-s + (−0.692 + 0.144i)10-s + (−0.141 − 0.618i)11-s + (−0.0109 − 0.0725i)12-s + (1.04 − 1.52i)13-s + (0.335 − 0.854i)14-s + (−0.146 − 0.0137i)15-s + (−0.225 − 0.108i)16-s + (−1.54 + 0.116i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.231 + 0.972i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.231 + 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05801 - 1.33898i\)
\(L(\frac12)\) \(\approx\) \(1.05801 - 1.33898i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.781 + 0.623i)T \)
5 \( 1 + (1.99 + 1.00i)T \)
43 \( 1 + (-3.32 - 5.65i)T \)
good3 \( 1 + (-0.236 + 0.0929i)T + (2.19 - 2.04i)T^{2} \)
7 \( 1 + (-2.97 + 1.71i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.468 + 2.05i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-3.75 + 5.50i)T + (-4.74 - 12.1i)T^{2} \)
17 \( 1 + (6.38 - 0.478i)T + (16.8 - 2.53i)T^{2} \)
19 \( 1 + (-3.12 - 2.90i)T + (1.41 + 18.9i)T^{2} \)
23 \( 1 + (-2.71 + 8.80i)T + (-19.0 - 12.9i)T^{2} \)
29 \( 1 + (1.36 - 3.47i)T + (-21.2 - 19.7i)T^{2} \)
31 \( 1 + (2.52 - 0.380i)T + (29.6 - 9.13i)T^{2} \)
37 \( 1 + (-1.10 - 0.637i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.65 - 2.07i)T + (-9.12 + 39.9i)T^{2} \)
47 \( 1 + (-11.4 - 2.61i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (1.05 + 1.55i)T + (-19.3 + 49.3i)T^{2} \)
59 \( 1 + (-5.38 - 2.59i)T + (36.7 + 46.1i)T^{2} \)
61 \( 1 + (-10.6 - 1.60i)T + (58.2 + 17.9i)T^{2} \)
67 \( 1 + (5.16 - 5.57i)T + (-5.00 - 66.8i)T^{2} \)
71 \( 1 + (9.49 - 2.92i)T + (58.6 - 39.9i)T^{2} \)
73 \( 1 + (1.03 - 1.52i)T + (-26.6 - 67.9i)T^{2} \)
79 \( 1 + (-0.212 - 0.367i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.12 - 1.62i)T + (60.8 - 56.4i)T^{2} \)
89 \( 1 + (-3.06 - 7.81i)T + (-65.2 + 60.5i)T^{2} \)
97 \( 1 + (-3.53 + 0.807i)T + (87.3 - 42.0i)T^{2} \)
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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

−10.84292174304214465279827769561, −10.70953858852722371389658016751, −8.698922555371547465450124204019, −8.319065464624881257419799163414, −7.37517536945402980434748137208, −5.84232072647748342202347450040, −4.91202339596713026921765460093, −3.99693153974478517251914173393, −2.78860903400737188021889759878, −0.965446134296536902682841026430, 2.21782037045119141967138143830, 3.67930043768624124561924312140, 4.53595914007377008163394424133, 5.69029083328036567839408449770, 6.84296349762567408058448582649, 7.55715444156790989485053423680, 8.782877470554041659522656335700, 9.113366752019523642378037454355, 11.10480986389938764278012910198, 11.48690997717917370685396368828

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