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Properties

Label 2-3332-476.447-c0-0-2
Degree $2$
Conductor $3332$
Sign $0.389 - 0.921i$
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 + (1.63 + 0.324i)5-s + (−0.923 − 0.382i)8-s + (0.382 − 0.923i)9-s + (0.324 + 1.63i)10-s + (1 − i)13-s − i·16-s + (−0.923 + 0.382i)17-s + 18-s + (−1.38 + 0.923i)20-s + (1.63 + 0.675i)25-s + (1.30 + 0.541i)26-s + (1.63 + 0.324i)29-s + (0.923 − 0.382i)32-s + ⋯
L(s)  = 1  + (0.382 + 0.923i)2-s + (−0.707 + 0.707i)4-s + (1.63 + 0.324i)5-s + (−0.923 − 0.382i)8-s + (0.382 − 0.923i)9-s + (0.324 + 1.63i)10-s + (1 − i)13-s − i·16-s + (−0.923 + 0.382i)17-s + 18-s + (−1.38 + 0.923i)20-s + (1.63 + 0.675i)25-s + (1.30 + 0.541i)26-s + (1.63 + 0.324i)29-s + (0.923 − 0.382i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.389 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.389 - 0.921i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.389 - 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.975873905\)
\(L(\frac12)\) \(\approx\) \(1.975873905\)
\(L(1)\) 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 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
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 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (1.92 - 0.382i)T + (0.923 - 0.382i)T^{2} \)
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 \( 1 + (-0.0761 - 0.382i)T + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.382 - 0.923i)T^{2} \)
73 \( 1 + (1.08 + 0.216i)T + (0.923 + 0.382i)T^{2} \)
79 \( 1 + (-0.382 - 0.923i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 + (0.216 - 1.08i)T + (-0.923 - 0.382i)T^{2} \)
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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

−8.805049128018332711871032151919, −8.347113881844791303147201627975, −7.09498430250001602953883493620, −6.57754371243510941595130139902, −6.00911076770880496784888614424, −5.45346891177713230317661247031, −4.48284874940336832496284783923, −3.48887547597368530991921859565, −2.68242428109627475267723014912, −1.29933171558254912411907023465, 1.43012057307323231097592152331, 1.97423523826817961495961025694, 2.81865317324375542194820081394, 4.06699859793128004887341224200, 4.89499945694014521726292285967, 5.34141496518693264879637340772, 6.40503321956872817354112170693, 6.75001581139574337802489940587, 8.435191657582294866053057667350, 8.733995591750073062314701886499

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