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Properties

Label 2-35-7.6-c4-0-7
Degree $2$
Conductor $35$
Sign $-0.999 - 0.0414i$
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  − 7.63·2-s − 12.6i·3-s + 42.2·4-s − 11.1i·5-s + 96.8i·6-s + (2.03 − 48.9i)7-s − 200.·8-s − 80.1·9-s + 85.3i·10-s − 105.·11-s − 536. i·12-s + 164. i·13-s + (−15.4 + 373. i)14-s − 141.·15-s + 852.·16-s + 59.1i·17-s + ⋯
L(s)  = 1  − 1.90·2-s − 1.41i·3-s + 2.63·4-s − 0.447i·5-s + 2.69i·6-s + (0.0414 − 0.999i)7-s − 3.12·8-s − 0.989·9-s + 0.853i·10-s − 0.872·11-s − 3.72i·12-s + 0.974i·13-s + (−0.0790 + 1.90i)14-s − 0.630·15-s + 3.32·16-s + 0.204i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0414i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.999 - 0.0414i$
Analytic conductor: \(3.61794\)
Root analytic conductor: \(1.90209\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :2),\ -0.999 - 0.0414i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.00838705 + 0.404606i\)
\(L(\frac12)\) \(\approx\) \(0.00838705 + 0.404606i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + 11.1iT \)
7 \( 1 + (-2.03 + 48.9i)T \)
good2 \( 1 + 7.63T + 16T^{2} \)
3 \( 1 + 12.6iT - 81T^{2} \)
11 \( 1 + 105.T + 1.46e4T^{2} \)
13 \( 1 - 164. iT - 2.85e4T^{2} \)
17 \( 1 - 59.1iT - 8.35e4T^{2} \)
19 \( 1 + 196. iT - 1.30e5T^{2} \)
23 \( 1 + 552.T + 2.79e5T^{2} \)
29 \( 1 - 633.T + 7.07e5T^{2} \)
31 \( 1 + 232. iT - 9.23e5T^{2} \)
37 \( 1 + 490.T + 1.87e6T^{2} \)
41 \( 1 - 249. iT - 2.82e6T^{2} \)
43 \( 1 - 1.56e3T + 3.41e6T^{2} \)
47 \( 1 + 2.79e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.70e3T + 7.89e6T^{2} \)
59 \( 1 + 6.38e3iT - 1.21e7T^{2} \)
61 \( 1 + 4.98e3iT - 1.38e7T^{2} \)
67 \( 1 + 6.68e3T + 2.01e7T^{2} \)
71 \( 1 - 6.30e3T + 2.54e7T^{2} \)
73 \( 1 - 1.81e3iT - 2.83e7T^{2} \)
79 \( 1 - 2.62e3T + 3.89e7T^{2} \)
83 \( 1 - 9.36e3iT - 4.74e7T^{2} \)
89 \( 1 + 4.39e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.58e4iT - 8.85e7T^{2} \)
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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

−15.81950112312517520987865044742, −13.81924658440212387118415523318, −12.43134889572070185979485456642, −11.23977622057503893797435022052, −9.939716269514374065169865179814, −8.416163138554453554328370098392, −7.51695635602999980201845025436, −6.54446165985989711574120538925, −1.94965141984434777110252163154, −0.49832222167753604881300073646, 2.78168034146106034682653541541, 5.78744403408162480107023918067, 7.83571419231222613800109412332, 8.982085574505964359107533891541, 10.12497641061768614876482532545, 10.67249595506251236703732215230, 12.04482709901618241613995885340, 14.89425274785369662860600137411, 15.69552979740080392210662750002, 16.24186340801651029332822788512

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