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Properties

Label 2-3332-476.219-c0-0-0
Degree $2$
Conductor $3332$
Sign $-0.992 + 0.123i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.241 + 1.83i)5-s + (−0.707 + 0.707i)8-s + (−0.258 − 0.965i)9-s + (−0.241 − 1.83i)10-s + 2i·13-s + (0.500 − 0.866i)16-s + (−0.965 − 0.258i)17-s + (0.499 + 0.866i)18-s + (0.707 + 1.70i)20-s + (−2.33 − 0.624i)25-s + (−0.517 − 1.93i)26-s + (−0.707 + 0.292i)29-s + (−0.258 + 0.965i)32-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.241 + 1.83i)5-s + (−0.707 + 0.707i)8-s + (−0.258 − 0.965i)9-s + (−0.241 − 1.83i)10-s + 2i·13-s + (0.500 − 0.866i)16-s + (−0.965 − 0.258i)17-s + (0.499 + 0.866i)18-s + (0.707 + 1.70i)20-s + (−2.33 − 0.624i)25-s + (−0.517 − 1.93i)26-s + (−0.707 + 0.292i)29-s + (−0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.992 + 0.123i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.992 + 0.123i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3826107989\)
\(L(\frac12)\) \(\approx\) \(0.3826107989\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.212814806429935721920708970787, −8.598054430484070434646093628469, −7.35601543031363373512742516932, −7.14173422237913466962827072261, −6.34186398396973886179809198030, −6.07045179121077950117500335395, −4.46253270700262539636371868341, −3.49869172565451016974559020648, −2.64217750023780438875714466586, −1.79088260996609759252790880776, 0.30312208883794207120868664705, 1.46801307288539836521436001588, 2.46942171609113272626444025911, 3.60273123175848643784808390330, 4.65326536293150269923649532673, 5.37770673173140194515710516939, 6.06952995640730482940340856199, 7.37741189956435490345072990307, 8.073167230864581439657226343279, 8.313424963349139841761222418618

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