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Properties

Label 2-58e2-116.67-c0-0-4
Degree $2$
Conductor $3364$
Sign $0.999 + 0.00261i$
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 + (−1.62 + 0.781i)5-s + (−0.974 − 0.222i)8-s + (0.222 − 0.974i)9-s + (−1.40 − 1.12i)10-s + (−0.0990 − 0.433i)13-s + (−0.222 − 0.974i)16-s − 1.24i·17-s + (0.974 − 0.222i)18-s + (0.400 − 1.75i)20-s + (1.40 − 1.75i)25-s + (0.347 − 0.277i)26-s + (0.781 − 0.623i)32-s + (1.12 − 0.541i)34-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (−0.623 + 0.781i)4-s + (−1.62 + 0.781i)5-s + (−0.974 − 0.222i)8-s + (0.222 − 0.974i)9-s + (−1.40 − 1.12i)10-s + (−0.0990 − 0.433i)13-s + (−0.222 − 0.974i)16-s − 1.24i·17-s + (0.974 − 0.222i)18-s + (0.400 − 1.75i)20-s + (1.40 − 1.75i)25-s + (0.347 − 0.277i)26-s + (0.781 − 0.623i)32-s + (1.12 − 0.541i)34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $0.999 + 0.00261i$
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: \((2,\ 3364,\ (\ :0),\ 0.999 + 0.00261i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7517882706\)
\(L(\frac12)\) \(\approx\) \(0.7517882706\)
\(L(1)\) 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 \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \)
7 \( 1 + (0.222 - 0.974i)T^{2} \)
11 \( 1 + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2} \)
17 \( 1 + 1.24iT - T^{2} \)
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 \( 1 + (-0.433 - 0.0990i)T + (0.900 + 0.433i)T^{2} \)
41 \( 1 + 0.445iT - T^{2} \)
43 \( 1 + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.974 + 0.777i)T + (0.222 - 0.974i)T^{2} \)
67 \( 1 + (0.900 + 0.433i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (-0.541 + 1.12i)T + (-0.623 - 0.781i)T^{2} \)
79 \( 1 + (-0.900 - 0.433i)T^{2} \)
83 \( 1 + (0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.781 + 1.62i)T + (-0.623 + 0.781i)T^{2} \)
97 \( 1 + (0.347 + 0.277i)T + (0.222 + 0.974i)T^{2} \)
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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

−8.549814982279750264679036191881, −7.76205344180992858778744371074, −7.31426485680792718476624892598, −6.71775228212666523399159297252, −5.97178768618168464220830508529, −4.84076676614358162106228339010, −4.20906832202077427941654265193, −3.36950751963046968860643793382, −2.90092970011938901808257011567, −0.44659596654065465258363951561, 1.18726022431312358912111132580, 2.26244850559362528556784009876, 3.51288000956145562580553789301, 4.08606010342060643044127036567, 4.73610095691319157779329510624, 5.37663744763047369690432546093, 6.50594639542404330429569322039, 7.52259018483026214626837279678, 8.263575498064712712422865456816, 8.636694860332529708169417703968

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