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Properties

Label 2-35-5.2-c2-0-2
Degree $2$
Conductor $35$
Sign $0.675 - 0.737i$
Analytic cond. $0.953680$
Root an. cond. $0.976565$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.85 + 1.85i)2-s + (0.163 − 0.163i)3-s + 2.90i·4-s + (−4.36 − 2.43i)5-s + 0.608·6-s + (−1.87 − 1.87i)7-s + (2.02 − 2.02i)8-s + 8.94i·9-s + (−3.58 − 12.6i)10-s − 1.59·11-s + (0.476 + 0.476i)12-s + (2.82 − 2.82i)13-s − 6.95i·14-s + (−1.11 + 0.315i)15-s + 19.1·16-s + (−11.1 − 11.1i)17-s + ⋯
L(s)  = 1  + (0.929 + 0.929i)2-s + (0.0546 − 0.0546i)3-s + 0.727i·4-s + (−0.873 − 0.487i)5-s + 0.101·6-s + (−0.267 − 0.267i)7-s + (0.253 − 0.253i)8-s + 0.994i·9-s + (−0.358 − 1.26i)10-s − 0.145·11-s + (0.0397 + 0.0397i)12-s + (0.217 − 0.217i)13-s − 0.496i·14-s + (−0.0742 + 0.0210i)15-s + 1.19·16-s + (−0.656 − 0.656i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.675 - 0.737i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.675 - 0.737i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.29158 + 0.568919i\)
\(L(\frac12)\) \(\approx\) \(1.29158 + 0.568919i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.36 + 2.43i)T \)
7 \( 1 + (1.87 + 1.87i)T \)
good2 \( 1 + (-1.85 - 1.85i)T + 4iT^{2} \)
3 \( 1 + (-0.163 + 0.163i)T - 9iT^{2} \)
11 \( 1 + 1.59T + 121T^{2} \)
13 \( 1 + (-2.82 + 2.82i)T - 169iT^{2} \)
17 \( 1 + (11.1 + 11.1i)T + 289iT^{2} \)
19 \( 1 - 21.3iT - 361T^{2} \)
23 \( 1 + (13.8 - 13.8i)T - 529iT^{2} \)
29 \( 1 + 15.9iT - 841T^{2} \)
31 \( 1 - 53.2T + 961T^{2} \)
37 \( 1 + (45.5 + 45.5i)T + 1.36e3iT^{2} \)
41 \( 1 + 65.5T + 1.68e3T^{2} \)
43 \( 1 + (-48.4 + 48.4i)T - 1.84e3iT^{2} \)
47 \( 1 + (-10.9 - 10.9i)T + 2.20e3iT^{2} \)
53 \( 1 + (-48.0 + 48.0i)T - 2.80e3iT^{2} \)
59 \( 1 - 53.4iT - 3.48e3T^{2} \)
61 \( 1 - 22.7T + 3.72e3T^{2} \)
67 \( 1 + (-4.35 - 4.35i)T + 4.48e3iT^{2} \)
71 \( 1 + 38.4T + 5.04e3T^{2} \)
73 \( 1 + (20.2 - 20.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 90.9iT - 6.24e3T^{2} \)
83 \( 1 + (74.1 - 74.1i)T - 6.88e3iT^{2} \)
89 \( 1 + 108. iT - 7.92e3T^{2} \)
97 \( 1 + (-48.6 - 48.6i)T + 9.40e3iT^{2} \)
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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

−16.03225181802603325411664644396, −15.55533837647218106127150806079, −14.04738849722409845090969585343, −13.26412945812633242202267823133, −11.98581292654887861174670493253, −10.34120070709057114428600683168, −8.245883038019311314802486021420, −7.16589926513167747324724864784, −5.42335129651082195441798379869, −4.05554113768480104668837002815, 3.04129271191453024624644169006, 4.39605003776870971783246722804, 6.59367473078566157042915168819, 8.516576773338927315097192717928, 10.39509192955935506682776845047, 11.59506193350479455970615073837, 12.34183936713912623068152843809, 13.58948198054047732534052007645, 14.87569608088726815315928432422, 15.73161639832904242565358781143

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