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Properties

Label 2-3332-3332.1495-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.801 + 0.598i$
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  + (0.955 − 0.294i)2-s + (0.266 + 0.680i)3-s + (0.826 − 0.563i)4-s + (0.455 + 0.571i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (0.341 − 0.317i)9-s + (−1.40 − 1.29i)11-s + (0.603 + 0.411i)12-s + (−0.425 + 1.86i)13-s + (0.365 − 0.930i)14-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.233 − 0.403i)18-s + (0.698 + 0.215i)21-s + (−1.72 − 0.829i)22-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)2-s + (0.266 + 0.680i)3-s + (0.826 − 0.563i)4-s + (0.455 + 0.571i)6-s + (0.623 − 0.781i)7-s + (0.623 − 0.781i)8-s + (0.341 − 0.317i)9-s + (−1.40 − 1.29i)11-s + (0.603 + 0.411i)12-s + (−0.425 + 1.86i)13-s + (0.365 − 0.930i)14-s + (0.365 − 0.930i)16-s + (0.0747 − 0.997i)17-s + (0.233 − 0.403i)18-s + (0.698 + 0.215i)21-s + (−1.72 − 0.829i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.801 + 0.598i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1495, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.801 + 0.598i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.620364540\)
\(L(\frac12)\) \(\approx\) \(2.620364540\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.955 + 0.294i)T \)
7 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (-0.0747 + 0.997i)T \)
good3 \( 1 + (-0.266 - 0.680i)T + (-0.733 + 0.680i)T^{2} \)
5 \( 1 + (-0.955 - 0.294i)T^{2} \)
11 \( 1 + (1.40 + 1.29i)T + (0.0747 + 0.997i)T^{2} \)
13 \( 1 + (0.425 - 1.86i)T + (-0.900 - 0.433i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.0332 + 0.443i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.365 - 0.930i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (-0.826 + 0.563i)T^{2} \)
53 \( 1 + (1.63 - 1.11i)T + (0.365 - 0.930i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (-0.365 - 0.930i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.78 - 0.858i)T + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.826 - 0.563i)T^{2} \)
79 \( 1 + (-0.733 - 1.26i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.535 - 0.496i)T + (0.0747 - 0.997i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.876694023626695280628366363633, −7.87795270680791736355432575844, −7.02970575319381856498580830922, −6.55397446265531249749594981385, −5.25161548501208230034663628553, −4.85654544998407510798183896170, −4.11342964533333174740743259444, −3.31719893358087850546903761491, −2.49656614761230653677973037554, −1.20486968366875468888792793000, 1.80198474382053369787654802187, 2.40181367480797648503921466942, 3.19075787722016316452044159530, 4.53396780064768752285200825279, 5.13793909460570075653949996881, 5.65056222054682167401128699347, 6.65081486020073554243824323791, 7.59832085548630627620107521337, 7.87937342393283821640909186136, 8.310031462084086188255966356080

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