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Properties

Label 2-3332-68.15-c0-0-0
Degree $2$
Conductor $3332$
Sign $-0.0758 + 0.997i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−1.70 + 0.707i)5-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−1.70 − 0.707i)10-s − 2i·13-s − 1.00·16-s + (−0.707 + 0.707i)17-s − 1.00·18-s + (−0.707 − 1.70i)20-s + (1.70 − 1.70i)25-s + (1.41 − 1.41i)26-s + (−0.707 + 0.292i)29-s + (−0.707 − 0.707i)32-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−1.70 + 0.707i)5-s + (−0.707 + 0.707i)8-s + (−0.707 + 0.707i)9-s + (−1.70 − 0.707i)10-s − 2i·13-s − 1.00·16-s + (−0.707 + 0.707i)17-s − 1.00·18-s + (−0.707 − 1.70i)20-s + (1.70 − 1.70i)25-s + (1.41 − 1.41i)26-s + (−0.707 + 0.292i)29-s + (−0.707 − 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0758 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.0758 + 0.997i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.0758 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09830779599\)
\(L(\frac12)\) \(\approx\) \(0.09830779599\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
7 \( 1 \)
17 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (0.707 + 0.707i)T^{2} \)
13 \( 1 + 2iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + (0.707 + 0.707i)T^{2} \)
29 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.707 - 0.707i)T^{2} \)
37 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1 + i)T + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.707 - 0.707i)T^{2} \)
73 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 + (0.707 + 0.707i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
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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

−8.873458225611108486354348159978, −8.216189498439473927835593291171, −7.79827373688586586370319895807, −7.30078587509890120538983113100, −6.35434157824868879582503173310, −5.57225774658692846766056837693, −4.79321831044558780763752905131, −3.88858241692880972587115339182, −3.25036334189636790909054403180, −2.56217090576691343655968275193, 0.04569502695715513764472224181, 1.46142493655953190919526617296, 2.77768634962451064038636191207, 3.67753770248503293400779734521, 4.34543523930817536884988244333, 4.73983473518180413407738537025, 5.87900314091183779137229109132, 6.75868738877614524214237089966, 7.37771749070740451089476889905, 8.488781967153261448908119588855

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