Properties

Label 2-35-35.24-c4-0-3
Degree $2$
Conductor $35$
Sign $0.109 - 0.994i$
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  + (−5.18 + 2.99i)2-s + (1.73 − 2.99i)3-s + (9.90 − 17.1i)4-s + (24.7 − 3.78i)5-s + 20.7i·6-s + (−18.5 + 45.3i)7-s + 22.8i·8-s + (34.5 + 59.7i)9-s + (−116. + 93.5i)10-s + (−90.7 + 157. i)11-s + (−34.2 − 59.3i)12-s + 258.·13-s + (−39.6 − 290. i)14-s + (31.4 − 80.6i)15-s + (90.1 + 156. i)16-s + (21.6 − 37.4i)17-s + ⋯
L(s)  = 1  + (−1.29 + 0.748i)2-s + (0.192 − 0.333i)3-s + (0.619 − 1.07i)4-s + (0.988 − 0.151i)5-s + 0.575i·6-s + (−0.378 + 0.925i)7-s + 0.356i·8-s + (0.426 + 0.737i)9-s + (−1.16 + 0.935i)10-s + (−0.749 + 1.29i)11-s + (−0.238 − 0.412i)12-s + 1.52·13-s + (−0.202 − 1.48i)14-s + (0.139 − 0.358i)15-s + (0.352 + 0.610i)16-s + (0.0749 − 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 - 0.994i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.109 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.109 - 0.994i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ 0.109 - 0.994i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.661452 + 0.592885i\)
\(L(\frac12)\) \(\approx\) \(0.661452 + 0.592885i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-24.7 + 3.78i)T \)
7 \( 1 + (18.5 - 45.3i)T \)
good2 \( 1 + (5.18 - 2.99i)T + (8 - 13.8i)T^{2} \)
3 \( 1 + (-1.73 + 2.99i)T + (-40.5 - 70.1i)T^{2} \)
11 \( 1 + (90.7 - 157. i)T + (-7.32e3 - 1.26e4i)T^{2} \)
13 \( 1 - 258.T + 2.85e4T^{2} \)
17 \( 1 + (-21.6 + 37.4i)T + (-4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-161. + 93.5i)T + (6.51e4 - 1.12e5i)T^{2} \)
23 \( 1 + (374. - 216. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + 647.T + 7.07e5T^{2} \)
31 \( 1 + (-540. - 312. i)T + (4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + (282. - 162. i)T + (9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 + 2.88e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.29e3iT - 3.41e6T^{2} \)
47 \( 1 + (177. + 307. i)T + (-2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-796. - 459. i)T + (3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (1.85e3 + 1.07e3i)T + (6.05e6 + 1.04e7i)T^{2} \)
61 \( 1 + (2.37e3 - 1.37e3i)T + (6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.44e3 - 1.98e3i)T + (1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 7.72e3T + 2.54e7T^{2} \)
73 \( 1 + (931. - 1.61e3i)T + (-1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (1.99e3 + 3.45e3i)T + (-1.94e7 + 3.37e7i)T^{2} \)
83 \( 1 + 6.81e3T + 4.74e7T^{2} \)
89 \( 1 + (1.33e3 - 769. i)T + (3.13e7 - 5.43e7i)T^{2} \)
97 \( 1 - 1.29e4T + 8.85e7T^{2} \)
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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

−16.07358884626267115503952941671, −15.46251624973997096921712500609, −13.66115809207845717054599140944, −12.64010516346836328331975065561, −10.47231179977716039451737141045, −9.491631486745879847992924372585, −8.387379367708766920832426389566, −7.04201781218030027814469902186, −5.62593756996445761574210158847, −1.87000871900329093673439762800, 1.03950460553327210735216199695, 3.30814972249656178380038006084, 6.19429556471060465078107292411, 8.174815961563337757572638369706, 9.418401323145485115663758905690, 10.31398801752012416329429222543, 11.15639943654266383984681359889, 13.09024723560038957385318115601, 14.10599890272437361846380490139, 16.00793050234671201318137702396

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