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Properties

Label 2-430-43.4-c1-0-0
Degree $2$
Conductor $430$
Sign $-0.851 - 0.525i$
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 + (−1.03 − 1.29i)3-s + (−0.222 − 0.974i)4-s + (0.900 + 0.433i)5-s + 1.65·6-s − 3.40·7-s + (0.900 + 0.433i)8-s + (0.0591 − 0.259i)9-s + (−0.900 + 0.433i)10-s + (−0.666 + 2.92i)11-s + (−1.03 + 1.29i)12-s + (−1.19 − 0.577i)13-s + (2.12 − 2.66i)14-s + (−0.367 − 1.61i)15-s + (−0.900 + 0.433i)16-s + (−1.28 + 0.618i)17-s + ⋯
L(s)  = 1  + (−0.440 + 0.552i)2-s + (−0.595 − 0.746i)3-s + (−0.111 − 0.487i)4-s + (0.402 + 0.194i)5-s + 0.675·6-s − 1.28·7-s + (0.318 + 0.153i)8-s + (0.0197 − 0.0863i)9-s + (−0.284 + 0.137i)10-s + (−0.200 + 0.880i)11-s + (−0.297 + 0.373i)12-s + (−0.332 − 0.160i)13-s + (0.567 − 0.711i)14-s + (−0.0950 − 0.416i)15-s + (−0.225 + 0.108i)16-s + (−0.311 + 0.149i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(430\)    =    \(2 \cdot 5 \cdot 43\)
Sign: $-0.851 - 0.525i$
Analytic conductor: \(3.43356\)
Root analytic conductor: \(1.85298\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{430} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 430,\ (\ :1/2),\ -0.851 - 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0656299 + 0.231395i\)
\(L(\frac12)\) \(\approx\) \(0.0656299 + 0.231395i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (6.34 + 1.65i)T \)
good3 \( 1 + (1.03 + 1.29i)T + (-0.667 + 2.92i)T^{2} \)
7 \( 1 + 3.40T + 7T^{2} \)
11 \( 1 + (0.666 - 2.92i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.19 + 0.577i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + (1.28 - 0.618i)T + (10.5 - 13.2i)T^{2} \)
19 \( 1 + (-0.589 - 2.58i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (1.38 - 6.06i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (0.0758 - 0.0950i)T + (-6.45 - 28.2i)T^{2} \)
31 \( 1 + (4.06 - 5.10i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + 7.10T + 37T^{2} \)
41 \( 1 + (0.325 - 0.408i)T + (-9.12 - 39.9i)T^{2} \)
47 \( 1 + (0.00664 + 0.0291i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (-4.64 + 2.23i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (0.907 - 0.436i)T + (36.7 - 46.1i)T^{2} \)
61 \( 1 + (-1.93 - 2.42i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (-0.693 - 3.04i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (1.41 + 6.20i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-2.38 - 1.14i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 - 5.72T + 79T^{2} \)
83 \( 1 + (5.94 + 7.45i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (7.62 + 9.56i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (0.537 - 2.35i)T + (-87.3 - 42.0i)T^{2} \)
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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

−11.63019859479582015338090856923, −10.29716131568315982912264497028, −9.800175361026967445194525504536, −8.892193121893564758885858773060, −7.45270437891557505535013207039, −6.92027811469068245263301514980, −6.12478074886950354518177995884, −5.26462489347180160714748656187, −3.50205720370871074948213861621, −1.75002433208044134343976871702, 0.18082266747862820689745201945, 2.46500739307984302624248163644, 3.70059878968912364315425938582, 4.89964381090404131516593355194, 5.96269292599380061054688210112, 6.96541160123968674393197770217, 8.370838634337945769537768820987, 9.303633423883667587925148147821, 9.965355476295929706894686265692, 10.66622025350777546704135526907

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