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Properties

Label 2-35-35.4-c3-0-3
Degree $2$
Conductor $35$
Sign $0.899 + 0.436i$
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  + (−3.52 + 2.03i)2-s + (−3.93 − 2.27i)3-s + (4.26 − 7.39i)4-s + (9.62 − 5.69i)5-s + 18.4·6-s + (17.0 + 7.19i)7-s + 2.17i·8-s + (−3.16 − 5.48i)9-s + (−22.3 + 39.6i)10-s + (24.5 − 42.4i)11-s + (−33.5 + 19.3i)12-s − 18.7i·13-s + (−74.7 + 9.36i)14-s + (−50.8 + 0.543i)15-s + (29.7 + 51.4i)16-s + (−61.0 − 35.2i)17-s + ⋯
L(s)  = 1  + (−1.24 + 0.718i)2-s + (−0.757 − 0.437i)3-s + (0.533 − 0.923i)4-s + (0.860 − 0.509i)5-s + 1.25·6-s + (0.921 + 0.388i)7-s + 0.0960i·8-s + (−0.117 − 0.203i)9-s + (−0.705 + 1.25i)10-s + (0.672 − 1.16i)11-s + (−0.808 + 0.466i)12-s − 0.399i·13-s + (−1.42 + 0.178i)14-s + (−0.874 + 0.00935i)15-s + (0.464 + 0.804i)16-s + (−0.871 − 0.503i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.899 + 0.436i$
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: \((2,\ 35,\ (\ :3/2),\ 0.899 + 0.436i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.616358 - 0.141580i\)
\(L(\frac12)\) \(\approx\) \(0.616358 - 0.141580i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-9.62 + 5.69i)T \)
7 \( 1 + (-17.0 - 7.19i)T \)
good2 \( 1 + (3.52 - 2.03i)T + (4 - 6.92i)T^{2} \)
3 \( 1 + (3.93 + 2.27i)T + (13.5 + 23.3i)T^{2} \)
11 \( 1 + (-24.5 + 42.4i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 18.7iT - 2.19e3T^{2} \)
17 \( 1 + (61.0 + 35.2i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-33.9 - 58.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-95.9 + 55.4i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 11.5T + 2.43e4T^{2} \)
31 \( 1 + (-47.1 + 81.6i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (73.5 - 42.4i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 465.T + 6.89e4T^{2} \)
43 \( 1 - 165. iT - 7.95e4T^{2} \)
47 \( 1 + (-164. + 94.7i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-536. - 309. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (215. - 372. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (104. + 181. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (581. + 335. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 808.T + 3.57e5T^{2} \)
73 \( 1 + (-90.3 - 52.1i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-634. - 1.09e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 738. iT - 5.71e5T^{2} \)
89 \( 1 + (227. + 393. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.11e3iT - 9.12e5T^{2} \)
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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

−16.64874661402460060951653124097, −15.12339485169948419713505362256, −13.65360191768570554768146503555, −12.10590117273416053925798199763, −10.83821083379821274083060145464, −9.218789952691175270958189994514, −8.376785456405820325116483638656, −6.63361302489908194018462651004, −5.54268736690480920805755528877, −1.02076002633747566114273876749, 1.86415910498411353857131335762, 4.99895808554948595513167567829, 7.05905891382657512250595366470, 8.891056254891177648581537541182, 10.09053946416319517097871309512, 10.91563835941103038945632084353, 11.75988065151440276857504228876, 13.75988451750009756856221710565, 15.05661467609123127629466333607, 16.94264107481819654995233925254

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