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Properties

Label 2-35-35.33-c1-0-0
Degree $2$
Conductor $35$
Sign $0.417 - 0.908i$
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  + (0.5 + 1.86i)2-s + (−1.86 − 0.5i)3-s + (−1.5 + 0.866i)4-s + (1.86 − 1.23i)5-s − 3.73i·6-s + (−2.5 − 0.866i)7-s + (0.366 + 0.366i)8-s + (0.633 + 0.366i)9-s + (3.23 + 2.86i)10-s + (−0.366 − 0.633i)11-s + (3.23 − 0.866i)12-s + (−2 + 2i)13-s + (0.366 − 5.09i)14-s + (−4.09 + 1.36i)15-s + (−2.23 + 3.86i)16-s + (0.267 − i)17-s + ⋯
L(s)  = 1  + (0.353 + 1.31i)2-s + (−1.07 − 0.288i)3-s + (−0.750 + 0.433i)4-s + (0.834 − 0.550i)5-s − 1.52i·6-s + (−0.944 − 0.327i)7-s + (0.129 + 0.129i)8-s + (0.211 + 0.122i)9-s + (1.02 + 0.906i)10-s + (−0.110 − 0.191i)11-s + (0.933 − 0.249i)12-s + (−0.554 + 0.554i)13-s + (0.0978 − 1.36i)14-s + (−1.05 + 0.352i)15-s + (−0.558 + 0.966i)16-s + (0.0649 − 0.242i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.417 - 0.908i$
Analytic conductor: \(0.279476\)
Root analytic conductor: \(0.528655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1/2),\ 0.417 - 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.586423 + 0.376105i\)
\(L(\frac12)\) \(\approx\) \(0.586423 + 0.376105i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.5 + 0.866i)T \)
good2 \( 1 + (-0.5 - 1.86i)T + (-1.73 + i)T^{2} \)
3 \( 1 + (1.86 + 0.5i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (0.366 + 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 2i)T - 13iT^{2} \)
17 \( 1 + (-0.267 + i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.36 - 2.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.96 + 1.86i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + (-0.464 + 0.267i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.26 + 4.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 0.464iT - 41T^{2} \)
43 \( 1 + (5.83 + 5.83i)T + 43iT^{2} \)
47 \( 1 + (-0.633 + 0.169i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (1.83 - 6.83i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.09 - 1.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.33 + 4.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.13 - 0.303i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.73T + 71T^{2} \)
73 \( 1 + (-3.46 - 0.928i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.83 + 3.36i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.09 + 3.09i)T - 83iT^{2} \)
89 \( 1 + (-8.33 + 14.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.92 - 7.92i)T + 97iT^{2} \)
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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

−16.87953017754252443824313521250, −16.04318428545722760705000887064, −14.50888365694594072447154882282, −13.36923635569328896728367211284, −12.35867052808607800355457882741, −10.64131038979286408187379748068, −9.016155391805571685312311114409, −7.02713094948501020345620315248, −6.12742042860604732342413657475, −5.02892437117064729044505224360, 2.87831884721365919588530969877, 5.15100185885921508299932878107, 6.64196071871561793304760132804, 9.605355614521322375568214136091, 10.44769742647340497977959157926, 11.39537090770584928811341159329, 12.60282445755374996299588715303, 13.46302558792333353113703021945, 15.18163927525137493244049735532, 16.64944998724582835828711567204

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