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Properties

Label 2-35-35.24-c2-0-5
Degree $2$
Conductor $35$
Sign $0.836 + 0.547i$
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  + (1.74 − 1.00i)2-s + (0.784 − 1.35i)3-s + (0.0184 − 0.0319i)4-s + (−3.93 + 3.08i)5-s − 3.15i·6-s + (−1.38 − 6.86i)7-s + 7.96i·8-s + (3.26 + 5.66i)9-s + (−3.74 + 9.32i)10-s + (4.78 − 8.29i)11-s + (−0.0289 − 0.0501i)12-s − 13.9·13-s + (−9.30 − 10.5i)14-s + (1.11 + 7.76i)15-s + (8.07 + 13.9i)16-s + (8.97 − 15.5i)17-s + ⋯
L(s)  = 1  + (0.870 − 0.502i)2-s + (0.261 − 0.453i)3-s + (0.00461 − 0.00798i)4-s + (−0.786 + 0.617i)5-s − 0.525i·6-s + (−0.198 − 0.980i)7-s + 0.995i·8-s + (0.363 + 0.628i)9-s + (−0.374 + 0.932i)10-s + (0.435 − 0.753i)11-s + (−0.00241 − 0.00417i)12-s − 1.07·13-s + (−0.664 − 0.753i)14-s + (0.0740 + 0.517i)15-s + (0.504 + 0.873i)16-s + (0.528 − 0.914i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.836 + 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.836 + 0.547i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.836 + 0.547i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.34691 - 0.401279i\)
\(L(\frac12)\) \(\approx\) \(1.34691 - 0.401279i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (3.93 - 3.08i)T \)
7 \( 1 + (1.38 + 6.86i)T \)
good2 \( 1 + (-1.74 + 1.00i)T + (2 - 3.46i)T^{2} \)
3 \( 1 + (-0.784 + 1.35i)T + (-4.5 - 7.79i)T^{2} \)
11 \( 1 + (-4.78 + 8.29i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + 13.9T + 169T^{2} \)
17 \( 1 + (-8.97 + 15.5i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (9.91 - 5.72i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-7.35 + 4.24i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 - 26.6T + 841T^{2} \)
31 \( 1 + (1.87 + 1.08i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (56.1 - 32.4i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 - 23.3iT - 1.68e3T^{2} \)
43 \( 1 - 0.506iT - 1.84e3T^{2} \)
47 \( 1 + (-34.3 - 59.4i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (47.7 + 27.5i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-54.5 - 31.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-43.9 + 25.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-33.7 - 19.4i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 3.47T + 5.04e3T^{2} \)
73 \( 1 + (-12.3 + 21.3i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (68.8 + 119. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 110.T + 6.88e3T^{2} \)
89 \( 1 + (-84.0 + 48.5i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 3.28T + 9.40e3T^{2} \)
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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

−16.17657752221031454631248650088, −14.44855126200937060826468934567, −13.86972564072923100047196390382, −12.66095655323236032185613630615, −11.58035830742928777551623399777, −10.38587434002223659101148975555, −8.139606025338066979341360485164, −7.03506896751846296019413218220, −4.58046948694508691385685060154, −3.06965339498103581628030859438, 3.91360885710228745106634561434, 5.17761713287380555749474736256, 6.91742494445114755689471176887, 8.831916313085302050897081986138, 9.945862610876311147528380130157, 12.23398222186424323368840673342, 12.61523034189808539044296835124, 14.51124519177190169238871591702, 15.21851709946105401835020989419, 15.83230865005992923630889667971

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