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Properties

Label 2-35-5.2-c4-0-9
Degree $2$
Conductor $35$
Sign $-0.452 + 0.891i$
Analytic cond. $3.61794$
Root an. cond. $1.90209$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.88 − 2.88i)2-s + (11.3 − 11.3i)3-s + 0.699i·4-s + (24.2 + 5.88i)5-s − 65.5·6-s + (13.0 + 13.0i)7-s + (−44.2 + 44.2i)8-s − 176. i·9-s + (−53.1 − 87.2i)10-s − 15.6·11-s + (7.93 + 7.93i)12-s + (−137. + 137. i)13-s − 75.6i·14-s + (342. − 208. i)15-s + 266.·16-s + (108. + 108. i)17-s + ⋯
L(s)  = 1  + (−0.722 − 0.722i)2-s + (1.26 − 1.26i)3-s + 0.0437i·4-s + (0.971 + 0.235i)5-s − 1.82·6-s + (0.267 + 0.267i)7-s + (−0.690 + 0.690i)8-s − 2.18i·9-s + (−0.531 − 0.872i)10-s − 0.129·11-s + (0.0551 + 0.0551i)12-s + (−0.815 + 0.815i)13-s − 0.386i·14-s + (1.52 − 0.928i)15-s + 1.04·16-s + (0.376 + 0.376i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.452 + 0.891i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ -0.452 + 0.891i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.825174 - 1.34417i\)
\(L(\frac12)\) \(\approx\) \(0.825174 - 1.34417i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-24.2 - 5.88i)T \)
7 \( 1 + (-13.0 - 13.0i)T \)
good2 \( 1 + (2.88 + 2.88i)T + 16iT^{2} \)
3 \( 1 + (-11.3 + 11.3i)T - 81iT^{2} \)
11 \( 1 + 15.6T + 1.46e4T^{2} \)
13 \( 1 + (137. - 137. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-108. - 108. i)T + 8.35e4iT^{2} \)
19 \( 1 - 375. iT - 1.30e5T^{2} \)
23 \( 1 + (-153. + 153. i)T - 2.79e5iT^{2} \)
29 \( 1 + 1.53e3iT - 7.07e5T^{2} \)
31 \( 1 + 201.T + 9.23e5T^{2} \)
37 \( 1 + (186. + 186. i)T + 1.87e6iT^{2} \)
41 \( 1 - 3.01e3T + 2.82e6T^{2} \)
43 \( 1 + (2.33e3 - 2.33e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (922. + 922. i)T + 4.87e6iT^{2} \)
53 \( 1 + (700. - 700. i)T - 7.89e6iT^{2} \)
59 \( 1 + 760. iT - 1.21e7T^{2} \)
61 \( 1 + 2.82e3T + 1.38e7T^{2} \)
67 \( 1 + (-2.58e3 - 2.58e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 5.87e3T + 2.54e7T^{2} \)
73 \( 1 + (3.61e3 - 3.61e3i)T - 2.83e7iT^{2} \)
79 \( 1 + 7.49e3iT - 3.89e7T^{2} \)
83 \( 1 + (2.53e3 - 2.53e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 2.54e3iT - 6.27e7T^{2} \)
97 \( 1 + (-7.25e3 - 7.25e3i)T + 8.85e7iT^{2} \)
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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

−14.72071086518698663228104055873, −14.28551004091455763369812522056, −12.97710761519241390216835165676, −11.79223663666418291809784140785, −9.993035523060527063965461667895, −9.080118810827414540795662795361, −7.84868507212655503008121974250, −6.17548402803849483436345669461, −2.58164465396497627241333359081, −1.59159757109771465128730764379, 3.00808876261988871109532090527, 5.05801372249510730107517753887, 7.40775744768025783240085210051, 8.712536662483302525461791062430, 9.488555281312335339359298626500, 10.46875100840268215011801823675, 12.93505457091232812844312771956, 14.22128672835917009355279384454, 15.14170677430257616322029978692, 16.16540233420091804124818970877

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