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Properties

Label 2-3332-476.291-c0-0-0
Degree $2$
Conductor $3332$
Sign $-0.513 - 0.857i$
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.465 + 0.607i)5-s + (0.707 − 0.707i)8-s + (−0.965 − 0.258i)9-s + (−0.465 − 0.607i)10-s − 2i·13-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)17-s + (0.499 − 0.866i)18-s + (0.707 − 0.292i)20-s + (0.107 + 0.400i)25-s + (1.93 + 0.517i)26-s + (0.707 + 1.70i)29-s + (−0.965 + 0.258i)32-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.465 + 0.607i)5-s + (0.707 − 0.707i)8-s + (−0.965 − 0.258i)9-s + (−0.465 − 0.607i)10-s − 2i·13-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)17-s + (0.499 − 0.866i)18-s + (0.707 − 0.292i)20-s + (0.107 + 0.400i)25-s + (1.93 + 0.517i)26-s + (0.707 + 1.70i)29-s + (−0.965 + 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.513 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.513 - 0.857i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.513 - 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7266444728\)
\(L(\frac12)\) \(\approx\) \(0.7266444728\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
7 \( 1 \)
17 \( 1 + (-0.258 - 0.965i)T \)
good3 \( 1 + (0.965 + 0.258i)T^{2} \)
5 \( 1 + (0.465 - 0.607i)T + (-0.258 - 0.965i)T^{2} \)
11 \( 1 + (-0.258 + 0.965i)T^{2} \)
13 \( 1 + 2iT - T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.965 - 0.258i)T^{2} \)
29 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 + (0.965 + 0.258i)T^{2} \)
37 \( 1 + (-1.46 - 1.12i)T + (0.258 + 0.965i)T^{2} \)
41 \( 1 + (0.707 - 1.70i)T + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (1.83 - 0.241i)T + (0.965 - 0.258i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.758 - 0.0999i)T + (0.965 + 0.258i)T^{2} \)
79 \( 1 + (0.965 - 0.258i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
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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

−8.721297099732867453834610963846, −8.176041226232431181798090771574, −7.73008590383441957500773571061, −6.81335732376923022784284275154, −6.09576630551531890900616614769, −5.49314348326376090580331220784, −4.69076839812427809893276012508, −3.42124044840710510385749605313, −3.00331807577184086874212112397, −1.08661392276431142217986290002, 0.57743158349372447365651190460, 2.03042327918176741754861910213, 2.71977948903573184038203509576, 3.96430985953028353199041307395, 4.42972531757706018779087759008, 5.25709719847957892882688235890, 6.25777628033056976022327901577, 7.34094383381050651405157367914, 8.021784841508155512364077114459, 8.835960812067333055289537133690

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