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Properties

Label 2-3381-483.482-c0-0-2
Degree $2$
Conductor $3381$
Sign $-0.852 - 0.522i$
Analytic cond. $1.68733$
Root an. cond. $1.29897$
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  + 1.41i·2-s + (−0.923 − 0.382i)3-s − 1.00·4-s + (0.541 − 1.30i)6-s + (0.707 + 0.707i)9-s + (0.923 + 0.382i)12-s − 0.765i·13-s − 0.999·16-s + (−1.00 + i)18-s + i·23-s + 25-s + 1.08·26-s + (−0.382 − 0.923i)27-s + 1.84i·31-s − 1.41i·32-s + ⋯
L(s)  = 1  + 1.41i·2-s + (−0.923 − 0.382i)3-s − 1.00·4-s + (0.541 − 1.30i)6-s + (0.707 + 0.707i)9-s + (0.923 + 0.382i)12-s − 0.765i·13-s − 0.999·16-s + (−1.00 + i)18-s + i·23-s + 25-s + 1.08·26-s + (−0.382 − 0.923i)27-s + 1.84i·31-s − 1.41i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 - 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.852 - 0.522i$
Analytic conductor: \(1.68733\)
Root analytic conductor: \(1.29897\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (3380, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3381,\ (\ :0),\ -0.852 - 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8172515268\)
\(L(\frac12)\) \(\approx\) \(0.8172515268\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 \)
23 \( 1 - iT \)
good2 \( 1 - 1.41iT - T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 0.765iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.84iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 0.765T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + 0.765T + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.84T + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 2iT - T^{2} \)
73 \( 1 - 1.84iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.710955512654305784915441650377, −8.158715974232498071525362694207, −7.32136714569935538606433795934, −6.90352097860721583743626204655, −6.20677631995573473853834223435, −5.26434259275643817419244860734, −5.20041803711694759177479649858, −3.99777855669928710228595404749, −2.66310641727241720546573265005, −1.23061491228854846766980341130, 0.61836218330807346035137174705, 1.80950611805973417983507192537, 2.78069483451100680291930176313, 3.85434487124736889529650254952, 4.39729197004560875336225918113, 5.15574764406709831110760988126, 6.28048386203210396085647178298, 6.74552094683577615656109793791, 7.79169028242271006159427578732, 8.995613589085301795641996667144

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