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Properties

Label 2-3332-476.311-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.663 - 0.748i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (0.608 − 0.793i)2-s + (−0.258 − 0.965i)4-s + (0.534 + 1.57i)5-s + (−0.923 − 0.382i)8-s + (−0.991 + 0.130i)9-s + (1.57 + 0.534i)10-s + (−1 + i)13-s + (−0.866 + 0.499i)16-s + (−0.130 + 0.991i)17-s + (−0.499 + 0.866i)18-s + (1.38 − 0.923i)20-s + (−1.40 + 1.07i)25-s + (0.184 + 1.40i)26-s + (1.63 + 0.324i)29-s + (−0.130 + 0.991i)32-s + ⋯
L(s)  = 1  + (0.608 − 0.793i)2-s + (−0.258 − 0.965i)4-s + (0.534 + 1.57i)5-s + (−0.923 − 0.382i)8-s + (−0.991 + 0.130i)9-s + (1.57 + 0.534i)10-s + (−1 + i)13-s + (−0.866 + 0.499i)16-s + (−0.130 + 0.991i)17-s + (−0.499 + 0.866i)18-s + (1.38 − 0.923i)20-s + (−1.40 + 1.07i)25-s + (0.184 + 1.40i)26-s + (1.63 + 0.324i)29-s + (−0.130 + 0.991i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.663 - 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.663 - 0.748i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (3167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.663 - 0.748i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.322646047\)
\(L(\frac12)\) \(\approx\) \(1.322646047\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.608 + 0.793i)T \)
7 \( 1 \)
17 \( 1 + (0.130 - 0.991i)T \)
good3 \( 1 + (0.991 - 0.130i)T^{2} \)
5 \( 1 + (-0.534 - 1.57i)T + (-0.793 + 0.608i)T^{2} \)
11 \( 1 + (-0.608 + 0.793i)T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + (-0.258 - 0.965i)T^{2} \)
23 \( 1 + (-0.991 - 0.130i)T^{2} \)
29 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (-0.991 + 0.130i)T^{2} \)
37 \( 1 + (0.491 - 0.996i)T + (-0.608 - 0.793i)T^{2} \)
41 \( 1 + (-1.92 + 0.382i)T + (0.923 - 0.382i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (1.83 + 0.241i)T + (0.965 + 0.258i)T^{2} \)
59 \( 1 + (0.258 - 0.965i)T^{2} \)
61 \( 1 + (0.293 - 0.257i)T + (0.130 - 0.991i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (0.732 - 0.835i)T + (-0.130 - 0.991i)T^{2} \)
79 \( 1 + (0.991 + 0.130i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (-0.866 - 0.5i)T^{2} \)
97 \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \)
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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

−9.175430773720130262630782107948, −8.250156086789296288085640488635, −7.14523907649296994017677319877, −6.42119760044200689669973488735, −6.01098545523467612072935482162, −5.01872849020710526683188079427, −4.14508349696832526930309718056, −3.07307609203687524867073333268, −2.62681371545503231488382208506, −1.77000008332226694898470075797, 0.60367659484654544932842997203, 2.40032011223918237020799365135, 3.16662460160596091032345210624, 4.56377832912830162424196794067, 4.86988060885505227118577981873, 5.67358649989842701492300967138, 6.12683356032777993109415737128, 7.29010344642362522736107913714, 7.998331199383159421327431725653, 8.569192454929786548545562874967

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