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Properties

Label 2-35-5.3-c2-0-1
Degree $2$
Conductor $35$
Sign $0.399 - 0.916i$
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  + (−0.950 + 0.950i)2-s + (0.269 + 0.269i)3-s + 2.19i·4-s + (4.00 + 2.99i)5-s − 0.512·6-s + (−1.87 + 1.87i)7-s + (−5.88 − 5.88i)8-s − 8.85i·9-s + (−6.65 + 0.954i)10-s + 16.9·11-s + (−0.591 + 0.591i)12-s + (−14.0 − 14.0i)13-s − 3.55i·14-s + (0.270 + 1.88i)15-s + 2.41·16-s + (1.45 − 1.45i)17-s + ⋯
L(s)  = 1  + (−0.475 + 0.475i)2-s + (0.0898 + 0.0898i)3-s + 0.548i·4-s + (0.800 + 0.599i)5-s − 0.0853·6-s + (−0.267 + 0.267i)7-s + (−0.735 − 0.735i)8-s − 0.983i·9-s + (−0.665 + 0.0954i)10-s + 1.54·11-s + (−0.0492 + 0.0492i)12-s + (−1.07 − 1.07i)13-s − 0.253i·14-s + (0.0180 + 0.125i)15-s + 0.150·16-s + (0.0858 − 0.0858i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.399 - 0.916i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.399 - 0.916i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.757607 + 0.496189i\)
\(L(\frac12)\) \(\approx\) \(0.757607 + 0.496189i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-4.00 - 2.99i)T \)
7 \( 1 + (1.87 - 1.87i)T \)
good2 \( 1 + (0.950 - 0.950i)T - 4iT^{2} \)
3 \( 1 + (-0.269 - 0.269i)T + 9iT^{2} \)
11 \( 1 - 16.9T + 121T^{2} \)
13 \( 1 + (14.0 + 14.0i)T + 169iT^{2} \)
17 \( 1 + (-1.45 + 1.45i)T - 289iT^{2} \)
19 \( 1 - 5.98iT - 361T^{2} \)
23 \( 1 + (-12.9 - 12.9i)T + 529iT^{2} \)
29 \( 1 + 25.1iT - 841T^{2} \)
31 \( 1 + 41.5T + 961T^{2} \)
37 \( 1 + (0.676 - 0.676i)T - 1.36e3iT^{2} \)
41 \( 1 + 36.8T + 1.68e3T^{2} \)
43 \( 1 + (-10.2 - 10.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (21.1 - 21.1i)T - 2.20e3iT^{2} \)
53 \( 1 + (21.6 + 21.6i)T + 2.80e3iT^{2} \)
59 \( 1 - 81.0iT - 3.48e3T^{2} \)
61 \( 1 + 58.7T + 3.72e3T^{2} \)
67 \( 1 + (18.7 - 18.7i)T - 4.48e3iT^{2} \)
71 \( 1 - 59.1T + 5.04e3T^{2} \)
73 \( 1 + (-57.8 - 57.8i)T + 5.32e3iT^{2} \)
79 \( 1 + 96.0iT - 6.24e3T^{2} \)
83 \( 1 + (34.7 + 34.7i)T + 6.88e3iT^{2} \)
89 \( 1 + 46.7iT - 7.92e3T^{2} \)
97 \( 1 + (-133. + 133. i)T - 9.40e3iT^{2} \)
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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

−16.92062738868482597935222856704, −15.30002196870911414638849269913, −14.51874854378854334028891524734, −12.87254471899224646710762332430, −11.79797161852824859356346536749, −9.822732722181738210090131879738, −9.063272709422196176916941739756, −7.26496337990669603423592913961, −6.12409355314624048965611016488, −3.33268292889253127847065153274, 1.82124124241106035048026440393, 4.97040337662895782144577002238, 6.70460619892135617638753136900, 8.915603274431803180937759969808, 9.676510627131910560709719511534, 10.98716688693121577226846848001, 12.35549567899797324929636190223, 13.87927219944867209523300284292, 14.60197325059511352967985691839, 16.58729569017320985752497682309

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