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Properties

Label 2-3332-476.331-c0-0-3
Degree $2$
Conductor $3332$
Sign $0.962 - 0.271i$
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 + (−1.46 + 1.12i)5-s + (−0.707 − 0.707i)8-s + (0.965 − 0.258i)9-s + (−1.46 − 1.12i)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 − 1.70i)20-s + (0.624 − 2.33i)25-s + (1.93 − 0.517i)26-s + (−0.707 − 0.292i)29-s + (0.965 + 0.258i)32-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−1.46 + 1.12i)5-s + (−0.707 − 0.707i)8-s + (0.965 − 0.258i)9-s + (−1.46 − 1.12i)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 − 1.70i)20-s + (0.624 − 2.33i)25-s + (1.93 − 0.517i)26-s + (−0.707 − 0.292i)29-s + (0.965 + 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.962 - 0.271i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8191737631\)
\(L(\frac12)\) \(\approx\) \(0.8191737631\)
\(L(1)\) 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 \( 1 + (1.46 - 1.12i)T + (0.258 - 0.965i)T^{2} \)
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 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
31 \( 1 + (-0.965 + 0.258i)T^{2} \)
37 \( 1 + (0.465 + 0.607i)T + (-0.258 + 0.965i)T^{2} \)
41 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
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 \( 1 + (0.0999 - 0.758i)T + (-0.965 - 0.258i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.241 + 1.83i)T + (-0.965 + 0.258i)T^{2} \)
79 \( 1 + (-0.965 - 0.258i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
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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

−8.460821619069396824450004124696, −7.77009642318559194963026993765, −7.38268317642748898704292462156, −6.87437439816688663166071003731, −5.91807914916219672274580449215, −5.06213924755462749641537326424, −4.14768212297803959342942287360, −3.46595150318003168243900735760, −2.86033546862184859570663045655, −0.52099088883558125701617172638, 1.24184580967821374918365488716, 1.97444567019157911734429425026, 3.63811592093273672741225142519, 4.01169757331518484525035510511, 4.63677126789104023780023226718, 5.30888868823132809437767433628, 6.60252488161988554987392414740, 7.40852773795306905196952405955, 8.283806123655658334670551238666, 8.828787554331482704421865163711

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