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Properties

Label 2-35-5.3-c4-0-0
Degree $2$
Conductor $35$
Sign $-0.756 - 0.654i$
Analytic cond. $3.61794$
Root an. cond. $1.90209$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.896 − 0.896i)2-s + (−9.60 − 9.60i)3-s + 14.3i·4-s + (−20.2 + 14.6i)5-s − 17.2·6-s + (−13.0 + 13.0i)7-s + (27.2 + 27.2i)8-s + 103. i·9-s + (−5.04 + 31.3i)10-s − 106.·11-s + (138. − 138. i)12-s + (−222. − 222. i)13-s + 23.4i·14-s + (335. + 54.0i)15-s − 181.·16-s + (163. − 163. i)17-s + ⋯
L(s)  = 1  + (0.224 − 0.224i)2-s + (−1.06 − 1.06i)3-s + 0.899i·4-s + (−0.810 + 0.585i)5-s − 0.478·6-s + (−0.267 + 0.267i)7-s + (0.425 + 0.425i)8-s + 1.27i·9-s + (−0.0504 + 0.313i)10-s − 0.879·11-s + (0.959 − 0.959i)12-s + (−1.31 − 1.31i)13-s + 0.119i·14-s + (1.49 + 0.240i)15-s − 0.708·16-s + (0.567 − 0.567i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.756 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.756 - 0.654i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ -0.756 - 0.654i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0627727 + 0.168467i\)
\(L(\frac12)\) \(\approx\) \(0.0627727 + 0.168467i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (20.2 - 14.6i)T \)
7 \( 1 + (13.0 - 13.0i)T \)
good2 \( 1 + (-0.896 + 0.896i)T - 16iT^{2} \)
3 \( 1 + (9.60 + 9.60i)T + 81iT^{2} \)
11 \( 1 + 106.T + 1.46e4T^{2} \)
13 \( 1 + (222. + 222. i)T + 2.85e4iT^{2} \)
17 \( 1 + (-163. + 163. i)T - 8.35e4iT^{2} \)
19 \( 1 - 406. iT - 1.30e5T^{2} \)
23 \( 1 + (-492. - 492. i)T + 2.79e5iT^{2} \)
29 \( 1 + 371. iT - 7.07e5T^{2} \)
31 \( 1 + 315.T + 9.23e5T^{2} \)
37 \( 1 + (1.11e3 - 1.11e3i)T - 1.87e6iT^{2} \)
41 \( 1 - 273.T + 2.82e6T^{2} \)
43 \( 1 + (739. + 739. i)T + 3.41e6iT^{2} \)
47 \( 1 + (-326. + 326. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-502. - 502. i)T + 7.89e6iT^{2} \)
59 \( 1 - 5.70e3iT - 1.21e7T^{2} \)
61 \( 1 + 974.T + 1.38e7T^{2} \)
67 \( 1 + (1.63e3 - 1.63e3i)T - 2.01e7iT^{2} \)
71 \( 1 - 586.T + 2.54e7T^{2} \)
73 \( 1 + (7.09e3 + 7.09e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 2.38e3iT - 3.89e7T^{2} \)
83 \( 1 + (1.29e3 + 1.29e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.14e4iT - 6.27e7T^{2} \)
97 \( 1 + (6.91e3 - 6.91e3i)T - 8.85e7iT^{2} \)
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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

−16.47730193119307435037530968408, −15.14833482732852612425762453482, −13.33029466219124571618879716485, −12.32902776625381303552172511947, −11.83696023048803268724172388000, −10.48386271795089467387369952091, −7.84482007351300525512937243361, −7.24390385140635872815207418370, −5.36011689983251050640466063280, −2.97440209417077186420760549669, 0.12828683269978223467612812089, 4.46899080750189141031527364757, 5.21702549480109479076303176541, 6.97803623725077063664549888528, 9.271690562582973996220570770072, 10.42661643096428544703548316786, 11.37969029396925106788226666232, 12.74161709559800316950779134227, 14.53540625186389563342092496107, 15.56046900063301052740239855443

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