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Properties

Label 2-435-29.25-c1-0-5
Degree $2$
Conductor $435$
Sign $0.165 - 0.986i$
Analytic cond. $3.47349$
Root an. cond. $1.86373$
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  + (−0.224 + 0.984i)2-s + (−0.900 + 0.433i)3-s + (0.883 + 0.425i)4-s + (0.222 − 0.974i)5-s + (−0.224 − 0.984i)6-s + (2.01 − 0.970i)7-s + (−1.87 + 2.35i)8-s + (0.623 − 0.781i)9-s + (0.909 + 0.438i)10-s + (3.66 + 4.59i)11-s − 0.980·12-s + (−3.64 − 4.56i)13-s + (0.502 + 2.20i)14-s + (0.222 + 0.974i)15-s + (−0.671 − 0.841i)16-s + 6.31·17-s + ⋯
L(s)  = 1  + (−0.158 + 0.695i)2-s + (−0.520 + 0.250i)3-s + (0.441 + 0.212i)4-s + (0.0995 − 0.436i)5-s + (−0.0917 − 0.401i)6-s + (0.761 − 0.366i)7-s + (−0.663 + 0.831i)8-s + (0.207 − 0.260i)9-s + (0.287 + 0.138i)10-s + (1.10 + 1.38i)11-s − 0.283·12-s + (−1.00 − 1.26i)13-s + (0.134 + 0.588i)14-s + (0.0574 + 0.251i)15-s + (−0.167 − 0.210i)16-s + 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $0.165 - 0.986i$
Analytic conductor: \(3.47349\)
Root analytic conductor: \(1.86373\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 435,\ (\ :1/2),\ 0.165 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06847 + 0.903937i\)
\(L(\frac12)\) \(\approx\) \(1.06847 + 0.903937i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.900 - 0.433i)T \)
5 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (0.912 - 5.30i)T \)
good2 \( 1 + (0.224 - 0.984i)T + (-1.80 - 0.867i)T^{2} \)
7 \( 1 + (-2.01 + 0.970i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-3.66 - 4.59i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (3.64 + 4.56i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 6.31T + 17T^{2} \)
19 \( 1 + (-1.99 - 0.962i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-0.662 - 2.90i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (0.939 - 4.11i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (-1.49 + 1.87i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + 9.94T + 41T^{2} \)
43 \( 1 + (0.664 + 2.91i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (3.80 + 4.77i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (0.497 - 2.17i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 12.6T + 59T^{2} \)
61 \( 1 + (-13.6 + 6.59i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (-2.55 + 3.19i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (8.21 + 10.2i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-1.64 - 7.20i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (-1.24 + 1.56i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (0.604 + 0.290i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-2.49 + 10.9i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (9.66 + 4.65i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.52197164776961858056194013891, −10.28751743127813228737544389797, −9.649481544632378884174635651284, −8.378232336780985908386667713347, −7.47019620270196139444138949483, −6.93812011697575506685543102046, −5.48403038812793549610452048195, −5.00017767219406492849622478246, −3.45917024135931421907112682928, −1.58247020584981938538222267788, 1.17684599147023541277410025023, 2.43457126896889218238745137190, 3.76964362509662216640252626777, 5.31628191514212102709681304468, 6.27950641520725637152006001894, 7.03337961475340549844762589230, 8.248657694135326305709100180236, 9.464974900747157290378780371723, 10.12554266313347727789881028022, 11.40366213291200010909783642089

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