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Properties

Label 2-35-7.6-c4-0-11
Degree $2$
Conductor $35$
Sign $-0.838 - 0.544i$
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  − 3.23·2-s − 14.6i·3-s − 5.54·4-s + 11.1i·5-s + 47.5i·6-s + (−26.6 + 41.0i)7-s + 69.6·8-s − 135.·9-s − 36.1i·10-s − 175.·11-s + 81.5i·12-s − 41.9i·13-s + (86.2 − 132. i)14-s + 164.·15-s − 136.·16-s + 11.9i·17-s + ⋯
L(s)  = 1  − 0.808·2-s − 1.63i·3-s − 0.346·4-s + 0.447i·5-s + 1.32i·6-s + (−0.544 + 0.838i)7-s + 1.08·8-s − 1.66·9-s − 0.361i·10-s − 1.44·11-s + 0.566i·12-s − 0.248i·13-s + (0.440 − 0.677i)14-s + 0.730·15-s − 0.533·16-s + 0.0415i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.838 - 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $-0.838 - 0.544i$
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: \((2,\ 35,\ (\ :2),\ -0.838 - 0.544i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0407514 + 0.137617i\)
\(L(\frac12)\) \(\approx\) \(0.0407514 + 0.137617i\)
\(L(3)\) 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 \( 1 + (26.6 - 41.0i)T \)
good2 \( 1 + 3.23T + 16T^{2} \)
3 \( 1 + 14.6iT - 81T^{2} \)
11 \( 1 + 175.T + 1.46e4T^{2} \)
13 \( 1 + 41.9iT - 2.85e4T^{2} \)
17 \( 1 - 11.9iT - 8.35e4T^{2} \)
19 \( 1 + 200. iT - 1.30e5T^{2} \)
23 \( 1 + 392.T + 2.79e5T^{2} \)
29 \( 1 + 1.37e3T + 7.07e5T^{2} \)
31 \( 1 + 1.58e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.90e3T + 1.87e6T^{2} \)
41 \( 1 - 2.43e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.04e3T + 3.41e6T^{2} \)
47 \( 1 + 2.54e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.28e3T + 7.89e6T^{2} \)
59 \( 1 - 1.91e3iT - 1.21e7T^{2} \)
61 \( 1 - 3.17e3iT - 1.38e7T^{2} \)
67 \( 1 - 6.95e3T + 2.01e7T^{2} \)
71 \( 1 + 5.47e3T + 2.54e7T^{2} \)
73 \( 1 + 358. iT - 2.83e7T^{2} \)
79 \( 1 - 342.T + 3.89e7T^{2} \)
83 \( 1 + 33.4iT - 4.74e7T^{2} \)
89 \( 1 - 81.3iT - 6.27e7T^{2} \)
97 \( 1 + 657. iT - 8.85e7T^{2} \)
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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

−15.06010238018369067958480866016, −13.35545957805378175560639768694, −12.98427552727954235890448395150, −11.40969674596691146701233711298, −9.796079488825321192225578758693, −8.279033717599109394596715546649, −7.41451125230141457353924378112, −5.81028752043229532831659817619, −2.35710335076475836682733424262, −0.12712121021265677767703431417, 3.89109306753690178997110248438, 5.17876984572570852307570623610, 7.86750805696958592655748434913, 9.216639896710638126600195659600, 10.09999201727759906141398724405, 10.81449229441440803935651671841, 13.00830727113211703160174454696, 14.25451067479931656491745994695, 15.85037048914841470147064916855, 16.39551708566780959965307046283

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