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Properties

Label 2-3332-476.139-c0-0-0
Degree $2$
Conductor $3332$
Sign $-0.570 - 0.820i$
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  + (−0.923 + 0.382i)2-s + (0.707 − 0.707i)4-s + (0.324 − 0.216i)5-s + (−0.382 + 0.923i)8-s + (−0.923 − 0.382i)9-s + (−0.216 + 0.324i)10-s + (−1 + i)13-s − i·16-s + (0.382 + 0.923i)17-s + 18-s + (0.0761 − 0.382i)20-s + (−0.324 + 0.783i)25-s + (0.541 − 1.30i)26-s + (−0.324 + 0.216i)29-s + (0.382 + 0.923i)32-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)2-s + (0.707 − 0.707i)4-s + (0.324 − 0.216i)5-s + (−0.382 + 0.923i)8-s + (−0.923 − 0.382i)9-s + (−0.216 + 0.324i)10-s + (−1 + i)13-s − i·16-s + (0.382 + 0.923i)17-s + 18-s + (0.0761 − 0.382i)20-s + (−0.324 + 0.783i)25-s + (0.541 − 1.30i)26-s + (−0.324 + 0.216i)29-s + (0.382 + 0.923i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $-0.570 - 0.820i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ -0.570 - 0.820i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4570380359\)
\(L(\frac12)\) \(\approx\) \(0.4570380359\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.923 - 0.382i)T \)
7 \( 1 \)
17 \( 1 + (-0.382 - 0.923i)T \)
good3 \( 1 + (0.923 + 0.382i)T^{2} \)
5 \( 1 + (-0.324 + 0.216i)T + (0.382 - 0.923i)T^{2} \)
11 \( 1 + (0.923 - 0.382i)T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + (0.707 - 0.707i)T^{2} \)
23 \( 1 + (-0.923 + 0.382i)T^{2} \)
29 \( 1 + (0.324 - 0.216i)T + (0.382 - 0.923i)T^{2} \)
31 \( 1 + (-0.923 - 0.382i)T^{2} \)
37 \( 1 + (1.92 + 0.382i)T + (0.923 + 0.382i)T^{2} \)
41 \( 1 + (-1.38 - 0.923i)T + (0.382 + 0.923i)T^{2} \)
43 \( 1 + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (0.617 - 0.923i)T + (-0.382 - 0.923i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.923 - 0.382i)T^{2} \)
73 \( 1 + (1.63 - 1.08i)T + (0.382 - 0.923i)T^{2} \)
79 \( 1 + (0.923 - 0.382i)T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (-1.08 - 1.63i)T + (-0.382 + 0.923i)T^{2} \)
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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

−9.076546363754481332859427693114, −8.461068250966191666386714341940, −7.58819793376078127658295681260, −6.98015004168892720751281188725, −6.07385649387455849445724317849, −5.59319404863814753206424202650, −4.65732308723604154785255643209, −3.40537319376834261299344118881, −2.33651516849284271809292187031, −1.43757215197905116700486468789, 0.35169130766545429036568927327, 1.98546723564408394121380620457, 2.75131713674030097519448378644, 3.40630547501026447624257053266, 4.79125151442604023245042200566, 5.61858089353086012755213700494, 6.39564889153704908392412907399, 7.45449487662231679895942417661, 7.73541175356816709009287910831, 8.674204149084784285894324916276

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