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Properties

Label 2-58e2-116.67-c0-0-3
Degree $2$
Conductor $3364$
Sign $-0.00261 + 0.999i$
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  + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.400 + 0.193i)5-s + (0.974 + 0.222i)8-s + (0.222 − 0.974i)9-s + (0.347 + 0.277i)10-s + (0.277 + 1.21i)13-s + (−0.222 − 0.974i)16-s − 1.80i·17-s + (−0.974 + 0.222i)18-s + (0.0990 − 0.433i)20-s + (−0.499 + 0.626i)25-s + (0.974 − 0.777i)26-s + (−0.781 + 0.623i)32-s + (−1.62 + 0.781i)34-s + ⋯
L(s)  = 1  + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (−0.400 + 0.193i)5-s + (0.974 + 0.222i)8-s + (0.222 − 0.974i)9-s + (0.347 + 0.277i)10-s + (0.277 + 1.21i)13-s + (−0.222 − 0.974i)16-s − 1.80i·17-s + (−0.974 + 0.222i)18-s + (0.0990 − 0.433i)20-s + (−0.499 + 0.626i)25-s + (0.974 − 0.777i)26-s + (−0.781 + 0.623i)32-s + (−1.62 + 0.781i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00261 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00261 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $-0.00261 + 0.999i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (1111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ -0.00261 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8507250237\)
\(L(\frac12)\) \(\approx\) \(0.8507250237\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.433 + 0.900i)T \)
29 \( 1 \)
good3 \( 1 + (-0.222 + 0.974i)T^{2} \)
5 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
7 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + 1.80iT - T^{2} \)
19 \( 1 + (-0.222 - 0.974i)T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.623 - 0.781i)T^{2} \)
37 \( 1 + (-1.21 - 0.277i)T + (0.900 + 0.433i)T^{2} \)
41 \( 1 + 1.24iT - T^{2} \)
43 \( 1 + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-1.40 + 1.12i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 + (0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.781 + 1.62i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 + (-0.900 - 0.433i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (-0.193 - 0.400i)T + (-0.623 + 0.781i)T^{2} \)
97 \( 1 + (0.974 + 0.777i)T + (0.222 + 0.974i)T^{2} \)
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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

−8.914673848985885892320310316039, −7.962743729205727294473888512420, −7.17076731457592890382495725334, −6.67942610755375093159500311463, −5.42276963542195863100090932582, −4.43301043719707298356306992464, −3.81110281226255530432212747253, −3.00192846021557925959723379518, −1.99024580252595664145964662638, −0.72673808369130747722708738746, 1.10934350144881783314137983788, 2.34875692221601904854297273801, 3.82079607723525310960983431428, 4.44129541341754940078596189666, 5.46005243737295075611181856486, 5.91605473758819369630189370155, 6.84017322878859131498919090082, 7.72719614799991374072511845049, 8.220751853875160245439402024428, 8.527830907193105649530669734745

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