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Properties

Label 2-58e2-116.23-c0-0-1
Degree $2$
Conductor $3364$
Sign $-0.661 - 0.750i$
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  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.277 + 0.347i)5-s + (0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.0990 − 0.433i)10-s + (−1.12 + 0.541i)13-s + (−0.900 + 0.433i)16-s + 1.80·17-s + (0.900 − 0.433i)18-s + (0.400 + 0.193i)20-s + (0.178 + 0.781i)25-s + (0.277 − 1.21i)26-s + (0.222 − 0.974i)32-s + (−1.12 + 1.40i)34-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)2-s + (−0.222 − 0.974i)4-s + (−0.277 + 0.347i)5-s + (0.900 + 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.0990 − 0.433i)10-s + (−1.12 + 0.541i)13-s + (−0.900 + 0.433i)16-s + 1.80·17-s + (0.900 − 0.433i)18-s + (0.400 + 0.193i)20-s + (0.178 + 0.781i)25-s + (0.277 − 1.21i)26-s + (0.222 − 0.974i)32-s + (−1.12 + 1.40i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $-0.661 - 0.750i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (1415, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ -0.661 - 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5749218222\)
\(L(\frac12)\) \(\approx\) \(0.5749218222\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 \)
good3 \( 1 + (0.900 + 0.433i)T^{2} \)
5 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
7 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (-0.623 + 0.781i)T^{2} \)
13 \( 1 + (1.12 - 0.541i)T + (0.623 - 0.781i)T^{2} \)
17 \( 1 - 1.80T + T^{2} \)
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 \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \)
41 \( 1 + 1.24T + T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.400 - 1.75i)T + (-0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 + (-0.623 - 0.781i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \)
97 \( 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2} \)
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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

−9.061663623740934709324101209555, −8.162138410066088491372893655931, −7.63740043219217732650317471512, −6.95758578554692281506104471483, −6.17530185610290082403071779086, −5.44697064829149563264350148396, −4.75003960557320234512567851971, −3.55038957972861752794926742976, −2.62474725246308002747744619856, −1.23454065814373724682000790844, 0.46025518223602622248531292202, 1.86847447225533733364266645653, 2.89914791786947313753157940101, 3.47050412990326815575698737374, 4.71478186358330631940875829627, 5.23670581703446892188787457000, 6.32621835303407124565642424649, 7.47692588764727791424395252301, 7.999553036803063206767290530395, 8.384302489807546738528015393877

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