Properties

Label 2-1323-21.20-c1-0-29
Degree $2$
Conductor $1323$
Sign $0.654 - 0.755i$
Analytic cond. $10.5642$
Root an. cond. $3.25026$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23i·2-s − 3.00·4-s − 2.23i·8-s − 4.47i·11-s − 1.73i·13-s − 0.999·16-s + 7.74·17-s − 3.46i·19-s + 10.0·22-s − 4.47i·23-s − 5·25-s + 3.87·26-s − 4.47i·29-s + 1.73i·31-s − 6.70i·32-s + ⋯
L(s)  = 1  + 1.58i·2-s − 1.50·4-s − 0.790i·8-s − 1.34i·11-s − 0.480i·13-s − 0.249·16-s + 1.87·17-s − 0.794i·19-s + 2.13·22-s − 0.932i·23-s − 25-s + 0.759·26-s − 0.830i·29-s + 0.311i·31-s − 1.18i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.654 - 0.755i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (1322, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.654 - 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.430042567\)
\(L(\frac12)\) \(\approx\) \(1.430042567\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 - 2.23iT - 2T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 + 4.47iT - 11T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 - 7.74T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 + 4.47iT - 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 - 5T + 37T^{2} \)
41 \( 1 - 7.74T + 41T^{2} \)
43 \( 1 + 7T + 43T^{2} \)
47 \( 1 - 7.74T + 47T^{2} \)
53 \( 1 + 4.47iT - 53T^{2} \)
59 \( 1 + 7.74T + 59T^{2} \)
61 \( 1 - 8.66iT - 61T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 - 8.94iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 7.74T + 83T^{2} \)
89 \( 1 - 15.4T + 89T^{2} \)
97 \( 1 - 1.73iT - 97T^{2} \)
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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

−9.443007943199987189007925497914, −8.608094823703619904613298671085, −7.948865322835162457559981845400, −7.42786188484917034396864864859, −6.26322123447981412029234839099, −5.83732176664004834820286838236, −5.05479678264218516576894645230, −3.92153750015967194636978542834, −2.77611866617057690361176747986, −0.67060593351823556159654370372, 1.28489050705477328282958369640, 2.10839342389064504094631454996, 3.30728838117662858687280142171, 4.03302284170833088873553239871, 4.98434096540528908124536930641, 6.01439227984217810795751816635, 7.32361781717128184921863007699, 7.930526298599680372931442056746, 9.285794193793045206104522208939, 9.675261515715116312941174797414

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