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Properties

Label 2-35-35.4-c3-0-8
Degree $2$
Conductor $35$
Sign $-0.904 + 0.425i$
Analytic cond. $2.06506$
Root an. cond. $1.43703$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.793 − 0.457i)2-s + (−7.59 − 4.38i)3-s + (−3.58 + 6.20i)4-s + (−1.99 − 11.0i)5-s − 8.03·6-s + (−13.8 − 12.2i)7-s + 13.8i·8-s + (24.9 + 43.2i)9-s + (−6.61 − 7.81i)10-s + (22.8 − 39.5i)11-s + (54.4 − 31.4i)12-s + 16.0i·13-s + (−16.6 − 3.36i)14-s + (−33.1 + 92.3i)15-s + (−22.2 − 38.5i)16-s + (−6.30 − 3.64i)17-s + ⋯
L(s)  = 1  + (0.280 − 0.161i)2-s + (−1.46 − 0.844i)3-s + (−0.447 + 0.775i)4-s + (−0.178 − 0.983i)5-s − 0.546·6-s + (−0.749 − 0.662i)7-s + 0.613i·8-s + (0.925 + 1.60i)9-s + (−0.209 − 0.247i)10-s + (0.625 − 1.08i)11-s + (1.30 − 0.755i)12-s + 0.343i·13-s + (−0.317 − 0.0643i)14-s + (−0.570 + 1.58i)15-s + (−0.348 − 0.603i)16-s + (−0.0899 − 0.0519i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 + 0.425i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.904 + 0.425i$
Analytic conductor: \(2.06506\)
Root analytic conductor: \(1.43703\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :3/2),\ -0.904 + 0.425i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.106476 - 0.476646i\)
\(L(\frac12)\) \(\approx\) \(0.106476 - 0.476646i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.99 + 11.0i)T \)
7 \( 1 + (13.8 + 12.2i)T \)
good2 \( 1 + (-0.793 + 0.457i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (7.59 + 4.38i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (-22.8 + 39.5i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 16.0iT - 2.19e3T^{2} \)
17 \( 1 + (6.30 + 3.64i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (37.8 + 65.4i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (13.7 - 7.93i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 119.T + 2.43e4T^{2} \)
31 \( 1 + (-29.7 + 51.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-263. + 152. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 238.T + 6.89e4T^{2} \)
43 \( 1 + 365. iT - 7.95e4T^{2} \)
47 \( 1 + (87.0 - 50.2i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-641. - 370. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (142. - 247. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (200. + 346. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-111. - 64.1i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 39.1T + 3.57e5T^{2} \)
73 \( 1 + (832. + 480. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (276. + 478. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 33.6iT - 5.71e5T^{2} \)
89 \( 1 + (428. + 741. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 771. iT - 9.12e5T^{2} \)
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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

−16.27343604234043728925028112125, −13.61957847409505022240669279407, −12.99390844293907102082361239313, −12.03449426154459866418814695615, −11.12181758235634937618289668700, −9.014300384166306080730377963134, −7.38516104955611654011301033115, −5.88601432767287873170309457091, −4.24320888039920204365481355118, −0.47470008871149197172773731416, 4.18561398794611906059357396271, 5.74944042885785986858846936247, 6.62069890857452638955631378718, 9.625068749752334824224021087224, 10.27586394306984644060738340974, 11.52896915133507461106641505605, 12.75191656504081715065494468414, 14.71649405548064971561685978181, 15.28171148865122779257316763540, 16.38202955443804001598688460467

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