Properties

Label 2-35-35.13-c3-0-9
Degree $2$
Conductor $35$
Sign $0.328 + 0.944i$
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  + (3.07 − 3.07i)2-s + (1.19 − 1.19i)3-s − 10.9i·4-s + (0.738 + 11.1i)5-s − 7.34i·6-s + (−13.5 − 12.6i)7-s + (−9.15 − 9.15i)8-s + 24.1i·9-s + (36.6 + 32.0i)10-s − 17.1·11-s + (−13.0 − 13.0i)12-s + (29.6 − 29.6i)13-s + (−80.6 + 2.98i)14-s + (14.1 + 12.4i)15-s + 31.3·16-s + (43.9 + 43.9i)17-s + ⋯
L(s)  = 1  + (1.08 − 1.08i)2-s + (0.229 − 0.229i)3-s − 1.37i·4-s + (0.0660 + 0.997i)5-s − 0.499i·6-s + (−0.732 − 0.680i)7-s + (−0.404 − 0.404i)8-s + 0.894i·9-s + (1.15 + 1.01i)10-s − 0.470·11-s + (−0.314 − 0.314i)12-s + (0.632 − 0.632i)13-s + (−1.53 + 0.0568i)14-s + (0.244 + 0.213i)15-s + 0.490·16-s + (0.626 + 0.626i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.328 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.328 + 0.944i$
Analytic conductor: \(2.06506\)
Root analytic conductor: \(1.43703\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :3/2),\ 0.328 + 0.944i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.69121 - 1.20213i\)
\(L(\frac12)\) \(\approx\) \(1.69121 - 1.20213i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.738 - 11.1i)T \)
7 \( 1 + (13.5 + 12.6i)T \)
good2 \( 1 + (-3.07 + 3.07i)T - 8iT^{2} \)
3 \( 1 + (-1.19 + 1.19i)T - 27iT^{2} \)
11 \( 1 + 17.1T + 1.33e3T^{2} \)
13 \( 1 + (-29.6 + 29.6i)T - 2.19e3iT^{2} \)
17 \( 1 + (-43.9 - 43.9i)T + 4.91e3iT^{2} \)
19 \( 1 + 93.7T + 6.85e3T^{2} \)
23 \( 1 + (90.3 + 90.3i)T + 1.21e4iT^{2} \)
29 \( 1 + 115. iT - 2.43e4T^{2} \)
31 \( 1 + 59.0iT - 2.97e4T^{2} \)
37 \( 1 + (65.2 - 65.2i)T - 5.06e4iT^{2} \)
41 \( 1 + 370. iT - 6.89e4T^{2} \)
43 \( 1 + (60.4 + 60.4i)T + 7.95e4iT^{2} \)
47 \( 1 + (-398. - 398. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-409. - 409. i)T + 1.48e5iT^{2} \)
59 \( 1 + 565.T + 2.05e5T^{2} \)
61 \( 1 - 677. iT - 2.26e5T^{2} \)
67 \( 1 + (-238. + 238. i)T - 3.00e5iT^{2} \)
71 \( 1 - 893.T + 3.57e5T^{2} \)
73 \( 1 + (-423. + 423. i)T - 3.89e5iT^{2} \)
79 \( 1 + 289. iT - 4.93e5T^{2} \)
83 \( 1 + (-453. + 453. i)T - 5.71e5iT^{2} \)
89 \( 1 + 514.T + 7.04e5T^{2} \)
97 \( 1 + (-44.1 - 44.1i)T + 9.12e5iT^{2} \)
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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

−15.39308168422357229711977161226, −14.09116276237784720819364677110, −13.42326382049269737266255927001, −12.41427471182296756272596545841, −10.61466103233613880850100444827, −10.49130441747711358266177136610, −7.81563800915285986989874207927, −6.00740789152380454613295164414, −3.92654567756476701621135955496, −2.49108598604886161901559043895, 3.78464622773287708575096923899, 5.37621219650018376279297700111, 6.55504796263378117453250976535, 8.398938740132002893715775388601, 9.622738912193563976921511479547, 12.08085047607090421347299170554, 12.91039711915485865954353511575, 13.99748469945129387326121037286, 15.27550570908378703008510994193, 15.97301750520281169736293255372

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