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Properties

Label 2-3332-476.335-c0-0-0
Degree $2$
Conductor $3332$
Sign $0.703 - 0.710i$
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  + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)4-s + (0.216 − 1.08i)5-s + (0.923 + 0.382i)8-s + (−0.382 + 0.923i)9-s + (−1.08 + 0.216i)10-s + (−1 + i)13-s − i·16-s + (−0.923 + 0.382i)17-s + 18-s + (0.617 + 0.923i)20-s + (−0.216 − 0.0897i)25-s + (1.30 + 0.541i)26-s + (−0.216 + 1.08i)29-s + (−0.923 + 0.382i)32-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)2-s + (−0.707 + 0.707i)4-s + (0.216 − 1.08i)5-s + (0.923 + 0.382i)8-s + (−0.382 + 0.923i)9-s + (−1.08 + 0.216i)10-s + (−1 + i)13-s − i·16-s + (−0.923 + 0.382i)17-s + 18-s + (0.617 + 0.923i)20-s + (−0.216 − 0.0897i)25-s + (1.30 + 0.541i)26-s + (−0.216 + 1.08i)29-s + (−0.923 + 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.703 - 0.710i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.703 - 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5165337323\)
\(L(\frac12)\) \(\approx\) \(0.5165337323\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 + 0.923i)T \)
7 \( 1 \)
17 \( 1 + (0.923 - 0.382i)T \)
good3 \( 1 + (0.382 - 0.923i)T^{2} \)
5 \( 1 + (-0.216 + 1.08i)T + (-0.923 - 0.382i)T^{2} \)
11 \( 1 + (0.382 + 0.923i)T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.382 - 0.923i)T^{2} \)
29 \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \)
31 \( 1 + (-0.382 + 0.923i)T^{2} \)
37 \( 1 + (1.38 - 0.923i)T + (0.382 - 0.923i)T^{2} \)
41 \( 1 + (-0.0761 - 0.382i)T + (-0.923 + 0.382i)T^{2} \)
43 \( 1 + (0.707 + 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.382 + 0.923i)T^{2} \)
73 \( 1 + (-0.324 + 1.63i)T + (-0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (1.63 + 0.324i)T + (0.923 + 0.382i)T^{2} \)
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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

−8.995875103157909156648558030439, −8.472475640258405937113208501865, −7.64225282545585870761945626612, −6.86559836332598586463346721844, −5.59590349200690638336022499284, −4.68484316696547438128147266509, −4.53660559270539343346669981378, −3.18059585951141172184598278916, −2.15756330160943100376000468340, −1.47972338541580235422754902440, 0.34204834707887138799949925468, 2.17677687779652228402065222307, 3.14119178614123357838605142706, 4.12488680894187369611997215341, 5.19964099480412437751180263359, 5.84611087122032345307195353670, 6.68057311485731805435311437747, 7.06198957331507211452827745007, 7.85236135718373616006071895425, 8.667756180408643802515503377018

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