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Properties

Label 2-35-35.32-c2-0-2
Degree $2$
Conductor $35$
Sign $0.998 - 0.0600i$
Analytic cond. $0.953680$
Root an. cond. $0.976565$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.330 − 1.23i)2-s + (1.07 + 4.01i)3-s + (2.05 + 1.18i)4-s + (−3.17 − 3.85i)5-s + 5.30·6-s + (−5.79 − 3.92i)7-s + (5.74 − 5.74i)8-s + (−7.17 + 4.14i)9-s + (−5.80 + 2.64i)10-s + (2.98 − 5.16i)11-s + (−2.55 + 9.53i)12-s + (−15.4 + 15.4i)13-s + (−6.74 + 5.84i)14-s + (12.0 − 16.9i)15-s + (−0.434 − 0.752i)16-s + (0.846 − 0.226i)17-s + ⋯
L(s)  = 1  + (0.165 − 0.615i)2-s + (0.358 + 1.33i)3-s + (0.513 + 0.296i)4-s + (−0.635 − 0.771i)5-s + 0.883·6-s + (−0.827 − 0.561i)7-s + (0.718 − 0.718i)8-s + (−0.797 + 0.460i)9-s + (−0.580 + 0.264i)10-s + (0.271 − 0.469i)11-s + (−0.212 + 0.794i)12-s + (−1.18 + 1.18i)13-s + (−0.482 + 0.417i)14-s + (0.805 − 1.12i)15-s + (−0.0271 − 0.0470i)16-s + (0.0498 − 0.0133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0600i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.998 - 0.0600i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.998 - 0.0600i$
Analytic conductor: \(0.953680\)
Root analytic conductor: \(0.976565\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :1),\ 0.998 - 0.0600i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.19930 + 0.0360135i\)
\(L(\frac12)\) \(\approx\) \(1.19930 + 0.0360135i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (3.17 + 3.85i)T \)
7 \( 1 + (5.79 + 3.92i)T \)
good2 \( 1 + (-0.330 + 1.23i)T + (-3.46 - 2i)T^{2} \)
3 \( 1 + (-1.07 - 4.01i)T + (-7.79 + 4.5i)T^{2} \)
11 \( 1 + (-2.98 + 5.16i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (15.4 - 15.4i)T - 169iT^{2} \)
17 \( 1 + (-0.846 + 0.226i)T + (250. - 144.5i)T^{2} \)
19 \( 1 + (-18.0 + 10.4i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (11.6 + 3.12i)T + (458. + 264.5i)T^{2} \)
29 \( 1 - 7.52iT - 841T^{2} \)
31 \( 1 + (9.90 - 17.1i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (3.59 - 13.4i)T + (-1.18e3 - 684.5i)T^{2} \)
41 \( 1 - 12.0T + 1.68e3T^{2} \)
43 \( 1 + (-12.3 + 12.3i)T - 1.84e3iT^{2} \)
47 \( 1 + (-11.7 + 43.9i)T + (-1.91e3 - 1.10e3i)T^{2} \)
53 \( 1 + (-2.62 - 9.78i)T + (-2.43e3 + 1.40e3i)T^{2} \)
59 \( 1 + (18.2 + 10.5i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (15.1 + 26.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-69.6 + 18.6i)T + (3.88e3 - 2.24e3i)T^{2} \)
71 \( 1 + 128.T + 5.04e3T^{2} \)
73 \( 1 + (19.4 + 72.5i)T + (-4.61e3 + 2.66e3i)T^{2} \)
79 \( 1 + (-14.8 + 8.57i)T + (3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-4.42 + 4.42i)T - 6.88e3iT^{2} \)
89 \( 1 + (94.1 - 54.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 + (55.4 + 55.4i)T + 9.40e3iT^{2} \)
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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

−16.35459605254607987994150079244, −15.49134763934812811401626703233, −13.94736634967894269940459758665, −12.46205820200792993069309280313, −11.42535387908750806000372818494, −10.08506112975955333556568439579, −9.093361744320307068047177201505, −7.21089740996769492053739927514, −4.52798478134377007115109536034, −3.41155186901682444421650361449, 2.63146120610269838771691985532, 5.93407062053864518617658146732, 7.18449417765807259078364624104, 7.80535294700074454299613427111, 10.05973760160091854689318569178, 11.79364298599219482513171073083, 12.69384936150432402356589129976, 14.17857708966086086585539924805, 15.05699684092191018652340185840, 16.02609515584110275864272219454

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