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Properties

Label 2-430-43.15-c1-0-1
Degree $2$
Conductor $430$
Sign $-0.781 - 0.623i$
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  + (0.623 − 0.781i)2-s + (−1.07 + 2.73i)3-s + (−0.222 − 0.974i)4-s + (−0.0747 − 0.997i)5-s + (1.46 + 2.54i)6-s + (−1.49 + 2.58i)7-s + (−0.900 − 0.433i)8-s + (−4.12 − 3.82i)9-s + (−0.826 − 0.563i)10-s + (−0.396 + 1.73i)11-s + (2.90 + 0.437i)12-s + (−1.88 + 1.28i)13-s + (1.09 + 2.77i)14-s + (2.80 + 0.865i)15-s + (−0.900 + 0.433i)16-s + (−0.0358 + 0.477i)17-s + ⋯
L(s)  = 1  + (0.440 − 0.552i)2-s + (−0.619 + 1.57i)3-s + (−0.111 − 0.487i)4-s + (−0.0334 − 0.445i)5-s + (0.599 + 1.03i)6-s + (−0.563 + 0.976i)7-s + (−0.318 − 0.153i)8-s + (−1.37 − 1.27i)9-s + (−0.261 − 0.178i)10-s + (−0.119 + 0.523i)11-s + (0.838 + 0.126i)12-s + (−0.523 + 0.356i)13-s + (0.291 + 0.742i)14-s + (0.724 + 0.223i)15-s + (−0.225 + 0.108i)16-s + (−0.00868 + 0.115i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.781 - 0.623i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.781 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.236211 + 0.674950i\)
\(L(\frac12)\) \(\approx\) \(0.236211 + 0.674950i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 + 0.781i)T \)
5 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (-3.29 - 5.67i)T \)
good3 \( 1 + (1.07 - 2.73i)T + (-2.19 - 2.04i)T^{2} \)
7 \( 1 + (1.49 - 2.58i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.396 - 1.73i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.88 - 1.28i)T + (4.74 - 12.1i)T^{2} \)
17 \( 1 + (0.0358 - 0.477i)T + (-16.8 - 2.53i)T^{2} \)
19 \( 1 + (1.79 - 1.66i)T + (1.41 - 18.9i)T^{2} \)
23 \( 1 + (7.96 - 2.45i)T + (19.0 - 12.9i)T^{2} \)
29 \( 1 + (2.89 + 7.36i)T + (-21.2 + 19.7i)T^{2} \)
31 \( 1 + (-4.28 - 0.645i)T + (29.6 + 9.13i)T^{2} \)
37 \( 1 + (-2.89 - 5.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.43 - 1.79i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (-0.301 - 1.32i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-9.63 - 6.56i)T + (19.3 + 49.3i)T^{2} \)
59 \( 1 + (-11.3 + 5.47i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (7.26 - 1.09i)T + (58.2 - 17.9i)T^{2} \)
67 \( 1 + (1.99 - 1.84i)T + (5.00 - 66.8i)T^{2} \)
71 \( 1 + (-5.37 - 1.65i)T + (58.6 + 39.9i)T^{2} \)
73 \( 1 + (-11.6 + 7.92i)T + (26.6 - 67.9i)T^{2} \)
79 \( 1 + (7.10 - 12.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.49 + 8.90i)T + (-60.8 - 56.4i)T^{2} \)
89 \( 1 + (3.38 - 8.62i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (0.0855 - 0.374i)T + (-87.3 - 42.0i)T^{2} \)
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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

−11.74796155959243468782683318448, −10.51301941052872859328181941523, −9.700579588809249026896216019006, −9.432428796832372942181984437139, −8.160927777268873098119541297856, −6.23939846309773232164821594385, −5.58558884760687161938978761982, −4.57820672248362942262266777613, −3.87992330084171167054505994324, −2.43595961011941664294816662721, 0.40033382359951051963161961180, 2.39419710034401106807298727057, 3.84809138770182854214349660494, 5.41430680606637096123778612218, 6.29857020743931011829427698847, 7.00845504978147888997691382888, 7.57529312133172508319919846384, 8.537414895854590812118717895819, 10.13876057664022776428474647496, 10.99637286440890263064623102938

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