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Properties

Label 2-35-5.3-c4-0-2
Degree $2$
Conductor $35$
Sign $0.828 - 0.559i$
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  + (−4.43 + 4.43i)2-s + (−10.8 − 10.8i)3-s − 23.3i·4-s + (8.86 + 23.3i)5-s + 95.9·6-s + (13.0 − 13.0i)7-s + (32.7 + 32.7i)8-s + 152. i·9-s + (−143. − 64.4i)10-s + 208.·11-s + (−252. + 252. i)12-s + (−1.92 − 1.92i)13-s + 116. i·14-s + (156. − 348. i)15-s + 83.2·16-s + (164. − 164. i)17-s + ⋯
L(s)  = 1  + (−1.10 + 1.10i)2-s + (−1.20 − 1.20i)3-s − 1.46i·4-s + (0.354 + 0.935i)5-s + 2.66·6-s + (0.267 − 0.267i)7-s + (0.512 + 0.512i)8-s + 1.88i·9-s + (−1.43 − 0.644i)10-s + 1.72·11-s + (−1.75 + 1.75i)12-s + (−0.0113 − 0.0113i)13-s + 0.593i·14-s + (0.697 − 1.54i)15-s + 0.325·16-s + (0.570 − 0.570i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.828 - 0.559i$
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: \((2,\ 35,\ (\ :2),\ 0.828 - 0.559i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.591708 + 0.181135i\)
\(L(\frac12)\) \(\approx\) \(0.591708 + 0.181135i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-8.86 - 23.3i)T \)
7 \( 1 + (-13.0 + 13.0i)T \)
good2 \( 1 + (4.43 - 4.43i)T - 16iT^{2} \)
3 \( 1 + (10.8 + 10.8i)T + 81iT^{2} \)
11 \( 1 - 208.T + 1.46e4T^{2} \)
13 \( 1 + (1.92 + 1.92i)T + 2.85e4iT^{2} \)
17 \( 1 + (-164. + 164. i)T - 8.35e4iT^{2} \)
19 \( 1 + 212. iT - 1.30e5T^{2} \)
23 \( 1 + (-391. - 391. i)T + 2.79e5iT^{2} \)
29 \( 1 + 36.5iT - 7.07e5T^{2} \)
31 \( 1 - 349.T + 9.23e5T^{2} \)
37 \( 1 + (-479. + 479. i)T - 1.87e6iT^{2} \)
41 \( 1 - 959.T + 2.82e6T^{2} \)
43 \( 1 + (-1.35e3 - 1.35e3i)T + 3.41e6iT^{2} \)
47 \( 1 + (-1.89e3 + 1.89e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-2.83e3 - 2.83e3i)T + 7.89e6iT^{2} \)
59 \( 1 + 3.14e3iT - 1.21e7T^{2} \)
61 \( 1 + 1.73e3T + 1.38e7T^{2} \)
67 \( 1 + (245. - 245. i)T - 2.01e7iT^{2} \)
71 \( 1 + 6.00e3T + 2.54e7T^{2} \)
73 \( 1 + (2.33e3 + 2.33e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 6.60e3iT - 3.89e7T^{2} \)
83 \( 1 + (-5.77e3 - 5.77e3i)T + 4.74e7iT^{2} \)
89 \( 1 - 1.07e4iT - 6.27e7T^{2} \)
97 \( 1 + (-3.20e3 + 3.20e3i)T - 8.85e7iT^{2} \)
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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

−16.49093176666063049500345573155, −14.88251284793604335183421634400, −13.73965597883393579487330721846, −11.97401818645237210146343510706, −10.90853605123843793988902280355, −9.359211968504906703878336189723, −7.47270419842772430802801097866, −6.81074250008367721966219586556, −5.83749370493561689976686544962, −1.07371651368124451939891947642, 1.07108400730343864428412998560, 4.15722477732642316191598478883, 5.84669865783392576177924102318, 8.741805381558026864023496895066, 9.562464962631477615455375970401, 10.57325505854440212322721998731, 11.73216096538951637081015913756, 12.34602250936664009605441393197, 14.72493330889576218586659589590, 16.39835619126166760926932175578

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