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Properties

Label 2-3332-3332.135-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.788 - 0.615i$
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  + (−0.826 + 0.563i)2-s + (−0.997 + 0.925i)3-s + (0.365 − 0.930i)4-s + (0.302 − 1.32i)6-s + (0.974 − 0.222i)7-s + (0.222 + 0.974i)8-s + (0.0635 − 0.848i)9-s + (−0.0841 − 1.12i)11-s + (0.496 + 1.26i)12-s + (1.48 + 0.716i)13-s + (−0.680 + 0.733i)14-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (0.425 + 0.736i)18-s + (−0.766 + 1.12i)21-s + (0.702 + 0.880i)22-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)2-s + (−0.997 + 0.925i)3-s + (0.365 − 0.930i)4-s + (0.302 − 1.32i)6-s + (0.974 − 0.222i)7-s + (0.222 + 0.974i)8-s + (0.0635 − 0.848i)9-s + (−0.0841 − 1.12i)11-s + (0.496 + 1.26i)12-s + (1.48 + 0.716i)13-s + (−0.680 + 0.733i)14-s + (−0.733 − 0.680i)16-s + (−0.988 − 0.149i)17-s + (0.425 + 0.736i)18-s + (−0.766 + 1.12i)21-s + (0.702 + 0.880i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.788 - 0.615i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.788 - 0.615i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6871667625\)
\(L(\frac12)\) \(\approx\) \(0.6871667625\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.826 - 0.563i)T \)
7 \( 1 + (-0.974 + 0.222i)T \)
17 \( 1 + (0.988 + 0.149i)T \)
good3 \( 1 + (0.997 - 0.925i)T + (0.0747 - 0.997i)T^{2} \)
5 \( 1 + (-0.826 - 0.563i)T^{2} \)
11 \( 1 + (0.0841 + 1.12i)T + (-0.988 + 0.149i)T^{2} \)
13 \( 1 + (-1.48 - 0.716i)T + (0.623 + 0.781i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.858 + 0.129i)T + (0.955 - 0.294i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.974 + 1.68i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.733 - 0.680i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.365 + 0.930i)T^{2} \)
53 \( 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2} \)
59 \( 1 + (-0.826 + 0.563i)T^{2} \)
61 \( 1 + (0.733 - 0.680i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.367 + 0.460i)T + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.365 - 0.930i)T^{2} \)
79 \( 1 + (-0.997 + 1.72i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (-0.109 + 1.46i)T + (-0.988 - 0.149i)T^{2} \)
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

−8.865147302321086401007807923064, −8.356687446892023947957248068065, −7.40878988605289469422696899339, −6.51279258841133665688157733622, −5.91868716172173508576143866859, −5.21810361843325993482173046195, −4.53834125789914560905395495492, −3.62659830537685171480210945874, −2.01107999415447734150929260852, −0.78403436064583527412418920883, 1.09895376787393161775256216249, 1.69633659702278923578541159927, 2.78718756952811722226016625516, 4.04412857388394656513096891107, 4.98538956853289925183770161165, 5.83346150391146038446882588573, 6.86117787210148242290715027570, 7.09489641885821884883580791834, 8.092596164706765649935201470398, 8.649059902684114477405934109052

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