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Properties

Label 2-58e2-116.103-c0-0-5
Degree $2$
Conductor $3364$
Sign $0.865 + 0.500i$
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  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−1.12 − 0.541i)5-s + (0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.777 − 0.974i)10-s + (0.400 − 1.75i)13-s + (−0.222 + 0.974i)16-s + 0.445·17-s + (0.222 − 0.974i)18-s + (−0.277 − 1.21i)20-s + (0.346 + 0.433i)25-s + (1.12 − 1.40i)26-s + (−0.623 + 0.781i)32-s + (0.400 + 0.193i)34-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−1.12 − 0.541i)5-s + (0.222 + 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.777 − 0.974i)10-s + (0.400 − 1.75i)13-s + (−0.222 + 0.974i)16-s + 0.445·17-s + (0.222 − 0.974i)18-s + (−0.277 − 1.21i)20-s + (0.346 + 0.433i)25-s + (1.12 − 1.40i)26-s + (−0.623 + 0.781i)32-s + (0.400 + 0.193i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $0.865 + 0.500i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (1031, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ 0.865 + 0.500i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.695516861\)
\(L(\frac12)\) \(\approx\) \(1.695516861\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 \)
good3 \( 1 + (0.222 + 0.974i)T^{2} \)
5 \( 1 + (1.12 + 0.541i)T + (0.623 + 0.781i)T^{2} \)
7 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (0.900 + 0.433i)T^{2} \)
13 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 - 0.445T + T^{2} \)
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 \( 1 + (0.400 + 1.75i)T + (-0.900 + 0.433i)T^{2} \)
41 \( 1 - 1.80T + T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \)
67 \( 1 + (0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.400 - 0.193i)T + (0.623 - 0.781i)T^{2} \)
79 \( 1 + (0.900 - 0.433i)T^{2} \)
83 \( 1 + (0.222 - 0.974i)T^{2} \)
89 \( 1 + (-1.12 - 0.541i)T + (0.623 + 0.781i)T^{2} \)
97 \( 1 + (-1.12 - 1.40i)T + (-0.222 + 0.974i)T^{2} \)
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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

−8.505156726710809942623442043903, −7.80047554453287540033867453890, −7.40428612626954692859038990467, −6.31681787624060556362988220197, −5.66436460097269941112285099510, −4.98096781644473641297981052062, −3.85036768697026978985439394499, −3.62074761548328297505923954455, −2.59568170186257787890233213897, −0.78332839472093545930299812141, 1.52348831398832598186238308732, 2.56316009992434839693099110045, 3.42195977342186002909625246952, 4.24796427800841437476411372982, 4.73083024264158658319194125274, 5.79782564506018993919398913789, 6.58809949938079739821598461297, 7.28322621216514031713517769259, 7.903952616214112480857127759961, 8.858551996239999550692596067356

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