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Properties

Label 2-35-7.5-c2-0-4
Degree $2$
Conductor $35$
Sign $0.676 + 0.736i$
Analytic cond. $0.953680$
Root an. cond. $0.976565$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.925 − 1.60i)2-s + (0.731 − 0.422i)3-s + (0.287 + 0.497i)4-s + (−1.93 − 1.11i)5-s − 1.56i·6-s + (−6.88 − 1.23i)7-s + 8.46·8-s + (−4.14 + 7.17i)9-s + (−3.58 + 2.06i)10-s + (3.93 + 6.82i)11-s + (0.420 + 0.242i)12-s − 2.73i·13-s + (−8.35 + 9.89i)14-s − 1.88·15-s + (6.68 − 11.5i)16-s + (4.29 − 2.47i)17-s + ⋯
L(s)  = 1  + (0.462 − 0.801i)2-s + (0.243 − 0.140i)3-s + (0.0718 + 0.124i)4-s + (−0.387 − 0.223i)5-s − 0.260i·6-s + (−0.984 − 0.176i)7-s + 1.05·8-s + (−0.460 + 0.797i)9-s + (−0.358 + 0.206i)10-s + (0.358 + 0.620i)11-s + (0.0350 + 0.0202i)12-s − 0.210i·13-s + (−0.597 + 0.706i)14-s − 0.125·15-s + (0.417 − 0.723i)16-s + (0.252 − 0.145i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (26, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.676 + 0.736i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.16999 - 0.514152i\)
\(L(\frac12)\) \(\approx\) \(1.16999 - 0.514152i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.93 + 1.11i)T \)
7 \( 1 + (6.88 + 1.23i)T \)
good2 \( 1 + (-0.925 + 1.60i)T + (-2 - 3.46i)T^{2} \)
3 \( 1 + (-0.731 + 0.422i)T + (4.5 - 7.79i)T^{2} \)
11 \( 1 + (-3.93 - 6.82i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + 2.73iT - 169T^{2} \)
17 \( 1 + (-4.29 + 2.47i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (25.3 + 14.6i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-18.0 + 31.3i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 - 0.399T + 841T^{2} \)
31 \( 1 + (-17.4 + 10.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (-3.61 + 6.25i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 71.1iT - 1.68e3T^{2} \)
43 \( 1 + 51.7T + 1.84e3T^{2} \)
47 \( 1 + (19.9 + 11.5i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-26.4 - 45.7i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-73.6 + 42.5i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-75.4 - 43.5i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (34.7 + 60.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 26.4T + 5.04e3T^{2} \)
73 \( 1 + (114. - 66.1i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (53.6 - 92.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + 53.9iT - 6.88e3T^{2} \)
89 \( 1 + (56.2 + 32.4i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 109. iT - 9.40e3T^{2} \)
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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

−16.33301740608193508661448792667, −14.81898108700627553977054469899, −13.30249144136510901505167596997, −12.70621470851392136472803493476, −11.41446426963522767234776856279, −10.21573069596827610762406118397, −8.438907922282186656122848408368, −6.91875745111200157559401878891, −4.48874981715854331686926267498, −2.74594152394609113460031098320, 3.66085429610605330003981008529, 5.85773713791331389985062891464, 6.88951853624010321191955419688, 8.656097309846567360707959148227, 10.17344442205414285669509620749, 11.70140199111484815515518359587, 13.19609617492018127071436014803, 14.44667856758290883207286144325, 15.22850080354919827033846588225, 16.19796978450223406044688288686

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