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Properties

Label 2-35-35.18-c2-0-3
Degree $2$
Conductor $35$
Sign $0.903 - 0.429i$
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  + (2.95 + 0.792i)2-s + (−2.36 + 0.633i)3-s + (4.66 + 2.69i)4-s + (−4.16 − 2.76i)5-s − 7.49·6-s + (5.78 − 3.94i)7-s + (2.99 + 2.99i)8-s + (−2.60 + 1.50i)9-s + (−10.1 − 11.4i)10-s + (−3.12 + 5.40i)11-s + (−12.7 − 3.40i)12-s + (11.7 + 11.7i)13-s + (20.2 − 7.07i)14-s + (11.6 + 3.88i)15-s + (−4.28 − 7.41i)16-s + (1.22 + 4.56i)17-s + ⋯
L(s)  = 1  + (1.47 + 0.396i)2-s + (−0.788 + 0.211i)3-s + (1.16 + 0.672i)4-s + (−0.833 − 0.552i)5-s − 1.24·6-s + (0.826 − 0.563i)7-s + (0.373 + 0.373i)8-s + (−0.289 + 0.167i)9-s + (−1.01 − 1.14i)10-s + (−0.283 + 0.491i)11-s + (−1.06 − 0.284i)12-s + (0.902 + 0.902i)13-s + (1.44 − 0.505i)14-s + (0.773 + 0.259i)15-s + (−0.267 − 0.463i)16-s + (0.0720 + 0.268i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.903 - 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.903 - 0.429i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.903 - 0.429i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.49269 + 0.336464i\)
\(L(\frac12)\) \(\approx\) \(1.49269 + 0.336464i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.16 + 2.76i)T \)
7 \( 1 + (-5.78 + 3.94i)T \)
good2 \( 1 + (-2.95 - 0.792i)T + (3.46 + 2i)T^{2} \)
3 \( 1 + (2.36 - 0.633i)T + (7.79 - 4.5i)T^{2} \)
11 \( 1 + (3.12 - 5.40i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (-11.7 - 11.7i)T + 169iT^{2} \)
17 \( 1 + (-1.22 - 4.56i)T + (-250. + 144.5i)T^{2} \)
19 \( 1 + (11.4 - 6.58i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-9.33 + 34.8i)T + (-458. - 264.5i)T^{2} \)
29 \( 1 - 28.5iT - 841T^{2} \)
31 \( 1 + (-10.1 + 17.6i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (-22.6 - 6.06i)T + (1.18e3 + 684.5i)T^{2} \)
41 \( 1 + 45.1T + 1.68e3T^{2} \)
43 \( 1 + (21.9 + 21.9i)T + 1.84e3iT^{2} \)
47 \( 1 + (10.4 + 2.79i)T + (1.91e3 + 1.10e3i)T^{2} \)
53 \( 1 + (71.6 - 19.2i)T + (2.43e3 - 1.40e3i)T^{2} \)
59 \( 1 + (14.6 + 8.43i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-16.7 - 29.0i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-25.6 - 95.8i)T + (-3.88e3 + 2.24e3i)T^{2} \)
71 \( 1 - 66.2T + 5.04e3T^{2} \)
73 \( 1 + (-99.8 + 26.7i)T + (4.61e3 - 2.66e3i)T^{2} \)
79 \( 1 + (36.1 - 20.8i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-7.39 - 7.39i)T + 6.88e3iT^{2} \)
89 \( 1 + (19.2 - 11.0i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (73.6 - 73.6i)T - 9.40e3iT^{2} \)
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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

−16.31666534400860516692599224768, −15.11084041322098711006277001939, −14.10877826251627596877456215729, −12.81792030156573883911642549392, −11.76889518015716785512677862653, −10.86210539747554996458187845914, −8.324853860378112743437667225022, −6.64650010453588751954610931663, −5.04607315275054768957670693963, −4.17038280724008634244566062206, 3.25695212097678383798826370518, 5.10524144638416598698915847172, 6.25531567090077105530126019277, 8.232504082038355079162570591697, 11.09039525977664879781484881799, 11.37997695237335519996569158154, 12.48728831992374638495886890294, 13.75957925019539895898466835001, 15.00822503069050100501260281043, 15.67394924866799380973668017239

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