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Properties

Label 2-58e2-116.71-c0-0-5
Degree $2$
Conductor $3364$
Sign $0.354 + 0.934i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 + 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3364\)    =    \(2^{2} \cdot 29^{2}\)
Sign: $0.354 + 0.934i$
Analytic conductor: \(1.67885\)
Root analytic conductor: \(1.29570\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3364} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3364,\ (\ :0),\ 0.354 + 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3640446779\)
\(L(\frac12)\) \(\approx\) \(0.3640446779\)
\(L(1)\) 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

−8.488538517601639198870950552108, −7.84696307306432616092557942174, −7.36892087501327380884214829338, −6.74699329801896956769363340906, −5.49577663605591502281712530634, −4.76269906501147594444998560166, −4.42388033450276833221715395189, −3.30968407040227482464851093041, −1.70844425349360091810439349902, −0.29456939687862371732963573272, 1.18293805394837599033848754942, 2.67683019283055385883663717870, 3.46519420895721447716586032361, 3.92947927598592384093239036030, 4.69720191953544939122321143846, 6.10991517259585369663690088085, 6.94968217154760264765485380396, 7.68719438214083422834947219291, 8.182259765892579807854263157245, 8.916546625044725605778307876403

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