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Properties

Label 2-35-5.4-c3-0-6
Degree $2$
Conductor $35$
Sign $0.936 + 0.350i$
Analytic cond. $2.06506$
Root an. cond. $1.43703$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.85i·2-s − 8.98i·3-s − 0.149·4-s + (3.91 − 10.4i)5-s + 25.6·6-s + 7i·7-s + 22.4i·8-s − 53.7·9-s + (29.8 + 11.1i)10-s + 37.4·11-s + 1.34i·12-s − 3.96i·13-s − 19.9·14-s + (−94.1 − 35.1i)15-s − 65.1·16-s + 51.6i·17-s + ⋯
L(s)  = 1  + 1.00i·2-s − 1.72i·3-s − 0.0186·4-s + (0.350 − 0.936i)5-s + 1.74·6-s + 0.377i·7-s + 0.990i·8-s − 1.99·9-s + (0.945 + 0.353i)10-s + 1.02·11-s + 0.0323i·12-s − 0.0845i·13-s − 0.381·14-s + (−1.62 − 0.605i)15-s − 1.01·16-s + 0.737i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(35\)    =    \(5 \cdot 7\)
Sign: $0.936 + 0.350i$
Analytic conductor: \(2.06506\)
Root analytic conductor: \(1.43703\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{35} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 35,\ (\ :3/2),\ 0.936 + 0.350i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.37680 - 0.248815i\)
\(L(\frac12)\) \(\approx\) \(1.37680 - 0.248815i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.91 + 10.4i)T \)
7 \( 1 - 7iT \)
good2 \( 1 - 2.85iT - 8T^{2} \)
3 \( 1 + 8.98iT - 27T^{2} \)
11 \( 1 - 37.4T + 1.33e3T^{2} \)
13 \( 1 + 3.96iT - 2.19e3T^{2} \)
17 \( 1 - 51.6iT - 4.91e3T^{2} \)
19 \( 1 + 25.9T + 6.85e3T^{2} \)
23 \( 1 - 173. iT - 1.21e4T^{2} \)
29 \( 1 - 245.T + 2.43e4T^{2} \)
31 \( 1 + 172.T + 2.97e4T^{2} \)
37 \( 1 + 250. iT - 5.06e4T^{2} \)
41 \( 1 + 48.8T + 6.89e4T^{2} \)
43 \( 1 - 143. iT - 7.95e4T^{2} \)
47 \( 1 + 36.6iT - 1.03e5T^{2} \)
53 \( 1 + 645. iT - 1.48e5T^{2} \)
59 \( 1 + 395.T + 2.05e5T^{2} \)
61 \( 1 - 47.5T + 2.26e5T^{2} \)
67 \( 1 - 263. iT - 3.00e5T^{2} \)
71 \( 1 + 268.T + 3.57e5T^{2} \)
73 \( 1 + 199. iT - 3.89e5T^{2} \)
79 \( 1 + 473.T + 4.93e5T^{2} \)
83 \( 1 - 72.7iT - 5.71e5T^{2} \)
89 \( 1 - 1.55e3T + 7.04e5T^{2} \)
97 \( 1 - 243. iT - 9.12e5T^{2} \)
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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

−16.19226496158436284458340390415, −14.67336157538052013347106803824, −13.63588328735384069902061337289, −12.52749823171013498060698068997, −11.59304126209408904767648962306, −8.914582456436068444159659952729, −7.909193556108086655952456576709, −6.62427752833086544809290263495, −5.64469708272380312089097318148, −1.71206031548227360801783289559, 2.98076640308672458033676084715, 4.30714501789098188166159705951, 6.55994999106142210774195079676, 9.182825031358158407379066449435, 10.23187880000332303284283992052, 10.83289901814441688068043380179, 11.92666315944695374324554192817, 13.97892688085318051312936060369, 14.93267070395509462269917137404, 16.06418001521645574326388405998

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