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Properties

Label 2-35-7.4-c1-0-1
Degree $2$
Conductor $35$
Sign $-0.198 + 0.980i$
Analytic cond. $0.279476$
Root an. cond. $0.528655$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 − 2.09i)2-s + (0.207 − 0.358i)3-s + (−1.91 + 3.31i)4-s + (−0.5 − 0.866i)5-s − 6-s + (2.62 + 0.358i)7-s + 4.41·8-s + (1.41 + 2.44i)9-s + (−1.20 + 2.09i)10-s + (0.414 − 0.717i)11-s + (0.792 + 1.37i)12-s − 4.82·13-s + (−2.41 − 5.91i)14-s − 0.414·15-s + (−1.49 − 2.59i)16-s + (−2.41 + 4.18i)17-s + ⋯
L(s)  = 1  + (−0.853 − 1.47i)2-s + (0.119 − 0.207i)3-s + (−0.957 + 1.65i)4-s + (−0.223 − 0.387i)5-s − 0.408·6-s + (0.990 + 0.135i)7-s + 1.56·8-s + (0.471 + 0.816i)9-s + (−0.381 + 0.661i)10-s + (0.124 − 0.216i)11-s + (0.228 + 0.396i)12-s − 1.33·13-s + (−0.645 − 1.58i)14-s − 0.106·15-s + (−0.374 − 0.649i)16-s + (−0.585 + 1.01i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(0.279476\)
Root analytic conductor: \(0.528655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332372 - 0.406249i\)
\(L(\frac12)\) \(\approx\) \(0.332372 - 0.406249i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.62 - 0.358i)T \)
good2 \( 1 + (1.20 + 2.09i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.207 + 0.358i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-0.414 + 0.717i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.82T + 13T^{2} \)
17 \( 1 + (2.41 - 4.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.207 + 0.358i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 - 3.58T + 43T^{2} \)
47 \( 1 + (1 + 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.585 + 1.01i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.24 - 3.88i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.74 + 4.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.79 - 8.30i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.48T + 71T^{2} \)
73 \( 1 + (-0.414 + 0.717i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.41 + 12.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 + (-4.32 - 7.49i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−16.89170553862582393734529839141, −15.05643379729391985440743834530, −13.38925708501227849862390675718, −12.31334337485115713826923228316, −11.26832080173664195164105518117, −10.20955880281173352222636554235, −8.763975966268183044027113514548, −7.74477049843809121485353588998, −4.55097300064152681458022245999, −2.08360306400790261273040738088, 4.84238648889317671558984205087, 6.74888324605567244040068521281, 7.71263000181428818990928551283, 9.094445240241736721926590931109, 10.22993831872043016925185227527, 11.99638652126369389252993355844, 14.16121949319904904757740790423, 14.90077895367364449460824843851, 15.69992099708420282197172482412, 17.02480970100776236278536054659

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