Properties

Label 2-58e2-116.23-c0-0-5
Degree $2$
Conductor $3364$
Sign $-0.853 + 0.521i$
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 + (0.777 − 0.974i)5-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.277 − 1.21i)10-s + (1.62 − 0.781i)13-s + (−0.900 + 0.433i)16-s − 0.445·17-s + (−0.900 + 0.433i)18-s + (−1.12 − 0.541i)20-s + (−0.123 − 0.541i)25-s + (0.400 − 1.75i)26-s + (−0.222 + 0.974i)32-s + (−0.277 + 0.347i)34-s + ⋯
L(s)  = 1  + (0.623 − 0.781i)2-s + (−0.222 − 0.974i)4-s + (0.777 − 0.974i)5-s + (−0.900 − 0.433i)8-s + (−0.900 − 0.433i)9-s + (−0.277 − 1.21i)10-s + (1.62 − 0.781i)13-s + (−0.900 + 0.433i)16-s − 0.445·17-s + (−0.900 + 0.433i)18-s + (−1.12 − 0.541i)20-s + (−0.123 − 0.541i)25-s + (0.400 − 1.75i)26-s + (−0.222 + 0.974i)32-s + (−0.277 + 0.347i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $-0.853 + 0.521i$
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: \((2,\ 3364,\ (\ :0),\ -0.853 + 0.521i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.802396689\)
\(L(\frac12)\) \(\approx\) \(1.802396689\)
\(L(1)\) 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 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
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 \( 1 + 0.445T + T^{2} \)
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 \( 1 + (-1.62 - 0.781i)T + (0.623 + 0.781i)T^{2} \)
41 \( 1 + 1.80T + T^{2} \)
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 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 + (-0.623 - 0.781i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.777 + 0.974i)T + (-0.222 - 0.974i)T^{2} \)
97 \( 1 + (-0.400 - 1.75i)T + (-0.900 + 0.433i)T^{2} \)
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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

−8.689399374803312428283901153333, −8.104579232618447923082322506364, −6.49391804007849976914136195841, −6.07073334827788455864257500275, −5.37893740987856483132596425278, −4.70986265014394613309595709112, −3.65478700234436113141541644284, −2.96894162155472768342237359063, −1.80030725281333404655264044284, −0.886895070456441456411712866099, 1.96867285347024772883967750196, 2.87492795964932702809552206742, 3.64260597683414204613203554132, 4.59523089867574250956592843294, 5.57053688753933778692280291558, 6.23592781709660231809453958417, 6.53639303189906664411549592518, 7.46223937006311593900704047127, 8.355213640672442485291148297593, 8.840209615014542794145175365638

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