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Properties

Label 2-35-35.17-c1-0-0
Degree $2$
Conductor $35$
Sign $0.908 - 0.417i$
Analytic cond. $0.279476$
Root an. cond. $0.528655$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.133 + 0.5i)2-s + (−0.5 + 0.133i)3-s + (1.5 + 0.866i)4-s + (−1.86 − 1.23i)5-s − 0.267i·6-s + (−0.866 − 2.5i)7-s + (−1.36 + 1.36i)8-s + (−2.36 + 1.36i)9-s + (0.866 − 0.767i)10-s + (1.36 − 2.36i)11-s + (−0.866 − 0.232i)12-s + (2 + 2i)13-s + (1.36 − 0.0980i)14-s + (1.09 + 0.366i)15-s + (1.23 + 2.13i)16-s + (1 + 3.73i)17-s + ⋯
L(s)  = 1  + (−0.0947 + 0.353i)2-s + (−0.288 + 0.0773i)3-s + (0.750 + 0.433i)4-s + (−0.834 − 0.550i)5-s − 0.109i·6-s + (−0.327 − 0.944i)7-s + (−0.482 + 0.482i)8-s + (−0.788 + 0.455i)9-s + (0.273 − 0.242i)10-s + (0.411 − 0.713i)11-s + (−0.249 − 0.0669i)12-s + (0.554 + 0.554i)13-s + (0.365 − 0.0262i)14-s + (0.283 + 0.0945i)15-s + (0.308 + 0.533i)16-s + (0.242 + 0.905i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 - 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.908 - 0.417i$
Analytic conductor: \(0.279476\)
Root analytic conductor: \(0.528655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1/2),\ 0.908 - 0.417i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.651935 + 0.142451i\)
\(L(\frac12)\) \(\approx\) \(0.651935 + 0.142451i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.86 + 1.23i)T \)
7 \( 1 + (0.866 + 2.5i)T \)
good2 \( 1 + (0.133 - 0.5i)T + (-1.73 - i)T^{2} \)
3 \( 1 + (0.5 - 0.133i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.36 + 2.36i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 2i)T + 13iT^{2} \)
17 \( 1 + (-1 - 3.73i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.366 + 0.633i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.133 + 0.0358i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (6.46 + 3.73i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.26 + 4.73i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.46iT - 41T^{2} \)
43 \( 1 + (-2.83 + 2.83i)T - 43iT^{2} \)
47 \( 1 + (-8.83 - 2.36i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.83 + 6.83i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.09 + 7.09i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.33 + 0.767i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.6 - 2.86i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.26T + 71T^{2} \)
73 \( 1 + (12.9 - 3.46i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.83 - 1.63i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.09 - 2.09i)T + 83iT^{2} \)
89 \( 1 + (-0.330 - 0.571i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.92 + 5.92i)T - 97iT^{2} \)
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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

−16.62029733954793546028210217758, −15.95705831945637399085029392698, −14.48512975802872504558708643680, −13.01297344342643144423827508327, −11.61051091263346178151558494604, −10.90174457918657298476398880026, −8.695421460037311522076147000745, −7.58251220231363947773738699121, −6.06833664267334239986107255110, −3.79965717230289315653433616725, 3.04912324141922401062734903461, 5.81916012346568186551336546357, 7.12719926899246584244220381514, 9.029323560353011140238202776180, 10.60003860974191964712564941436, 11.72109912650445411801721735731, 12.32704359946299549833186944525, 14.54502657986355296273577464670, 15.33933094952819773300114532837, 16.27853053464759205242710175447

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