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Properties

Label 2-58e2-116.35-c0-0-0
Degree $2$
Conductor $3364$
Sign $-0.669 - 0.742i$
Analytic cond. $1.67885$
Root an. cond. $1.29570$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.781 − 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.777 + 0.974i)5-s + (0.433 − 0.900i)8-s + (0.900 + 0.433i)9-s + (1.21 − 0.277i)10-s + (−1.62 + 0.781i)13-s + (−0.900 + 0.433i)16-s + 0.445i·17-s + (−0.433 − 0.900i)18-s + (−1.12 − 0.541i)20-s + (−0.123 − 0.541i)25-s + (1.75 + 0.400i)26-s + (0.974 + 0.222i)32-s + (0.277 − 0.347i)34-s + ⋯
L(s)  = 1  + (−0.781 − 0.623i)2-s + (0.222 + 0.974i)4-s + (−0.777 + 0.974i)5-s + (0.433 − 0.900i)8-s + (0.900 + 0.433i)9-s + (1.21 − 0.277i)10-s + (−1.62 + 0.781i)13-s + (−0.900 + 0.433i)16-s + 0.445i·17-s + (−0.433 − 0.900i)18-s + (−1.12 − 0.541i)20-s + (−0.123 − 0.541i)25-s + (1.75 + 0.400i)26-s + (0.974 + 0.222i)32-s + (0.277 − 0.347i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $-0.669 - 0.742i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ -0.669 - 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3804744520\)
\(L(\frac12)\) \(\approx\) \(0.3804744520\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.781 + 0.623i)T \)
29 \( 1 \)
good3 \( 1 + (-0.900 - 0.433i)T^{2} \)
5 \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \)
7 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.623 - 0.781i)T^{2} \)
13 \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 - 0.445iT - T^{2} \)
19 \( 1 + (-0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (-0.222 - 0.974i)T^{2} \)
37 \( 1 + (0.781 - 1.62i)T + (-0.623 - 0.781i)T^{2} \)
41 \( 1 + 1.80iT - T^{2} \)
43 \( 1 + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.433 + 0.0990i)T + (0.900 + 0.433i)T^{2} \)
67 \( 1 + (-0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.347 + 0.277i)T + (0.222 - 0.974i)T^{2} \)
79 \( 1 + (0.623 + 0.781i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.974 + 0.777i)T + (0.222 + 0.974i)T^{2} \)
97 \( 1 + (1.75 - 0.400i)T + (0.900 - 0.433i)T^{2} \)
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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

−9.196647937263789477987269923755, −8.256581724768069446040089221090, −7.56433332004915034542788794880, −7.07998646958899714620044887327, −6.58146492896929387798971295837, −5.00045936415820115768697453414, −4.22726164987061492935118873106, −3.44128942878361917274088226875, −2.51817039140441030713360112351, −1.67482149566391012260000852443, 0.29802939374844791930824636087, 1.45331100129492117104304557006, 2.75493365593327386538682049154, 4.12573280419249973034387897628, 4.85350288563457316196730049472, 5.39815243927985631186555569512, 6.53196824557712261210089265452, 7.27585446669135873300243921249, 7.78851992852988451510175255511, 8.374083009105018769941509273889

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