Properties

Label 2-35-5.3-c4-0-7
Degree $2$
Conductor $35$
Sign $-0.245 + 0.969i$
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  + (−2.97 + 2.97i)2-s + (−1.22 − 1.22i)3-s − 1.69i·4-s + (−22.1 − 11.5i)5-s + 7.31·6-s + (13.0 − 13.0i)7-s + (−42.5 − 42.5i)8-s − 77.9i·9-s + (100. − 31.6i)10-s + 17.6·11-s + (−2.07 + 2.07i)12-s + (−160. − 160. i)13-s + 77.8i·14-s + (13.0 + 41.4i)15-s + 280.·16-s + (−324. + 324. i)17-s + ⋯
L(s)  = 1  + (−0.743 + 0.743i)2-s + (−0.136 − 0.136i)3-s − 0.105i·4-s + (−0.887 − 0.461i)5-s + 0.203·6-s + (0.267 − 0.267i)7-s + (−0.664 − 0.664i)8-s − 0.962i·9-s + (1.00 − 0.316i)10-s + 0.145·11-s + (−0.0144 + 0.0144i)12-s + (−0.949 − 0.949i)13-s + 0.397i·14-s + (0.0581 + 0.184i)15-s + 1.09·16-s + (−1.12 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.245 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.245 + 0.969i$
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: \((2,\ 35,\ (\ :2),\ -0.245 + 0.969i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.171381 - 0.220213i\)
\(L(\frac12)\) \(\approx\) \(0.171381 - 0.220213i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (22.1 + 11.5i)T \)
7 \( 1 + (-13.0 + 13.0i)T \)
good2 \( 1 + (2.97 - 2.97i)T - 16iT^{2} \)
3 \( 1 + (1.22 + 1.22i)T + 81iT^{2} \)
11 \( 1 - 17.6T + 1.46e4T^{2} \)
13 \( 1 + (160. + 160. i)T + 2.85e4iT^{2} \)
17 \( 1 + (324. - 324. i)T - 8.35e4iT^{2} \)
19 \( 1 - 468. iT - 1.30e5T^{2} \)
23 \( 1 + (625. + 625. i)T + 2.79e5iT^{2} \)
29 \( 1 + 755. iT - 7.07e5T^{2} \)
31 \( 1 - 553.T + 9.23e5T^{2} \)
37 \( 1 + (-555. + 555. i)T - 1.87e6iT^{2} \)
41 \( 1 + 1.68e3T + 2.82e6T^{2} \)
43 \( 1 + (416. + 416. i)T + 3.41e6iT^{2} \)
47 \( 1 + (319. - 319. i)T - 4.87e6iT^{2} \)
53 \( 1 + (-3.35e3 - 3.35e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 4.67e3iT - 1.21e7T^{2} \)
61 \( 1 - 848.T + 1.38e7T^{2} \)
67 \( 1 + (-2.47e3 + 2.47e3i)T - 2.01e7iT^{2} \)
71 \( 1 + 2.45e3T + 2.54e7T^{2} \)
73 \( 1 + (2.34e3 + 2.34e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 1.76e3iT - 3.89e7T^{2} \)
83 \( 1 + (885. + 885. i)T + 4.74e7iT^{2} \)
89 \( 1 - 3.51e3iT - 6.27e7T^{2} \)
97 \( 1 + (-421. + 421. i)T - 8.85e7iT^{2} \)
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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

−15.55626288145457611131856494741, −14.80844390128628729232829843656, −12.69606272981120217779095191212, −11.98588142666724247474194832048, −10.12639389228431795656759355629, −8.580928546412894993593974202187, −7.74963407457418344147490351055, −6.30450104464202513401945755444, −3.97648084660056758964217946186, −0.24389848267706104084855796792, 2.39684078814528806036840330777, 4.83747352752917552830315097755, 7.14565299041767772032382781919, 8.679082741069799225717265600368, 9.979000796722858180062149636127, 11.36243225777173653041993399546, 11.69713259332858045782986928164, 13.78857513995934216651207045555, 15.05938413252424937518409238427, 16.15218824505924921314590598497

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