Properties

Label 2-1323-63.41-c1-0-29
Degree $2$
Conductor $1323$
Sign $0.424 + 0.905i$
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  + (1.02 − 0.589i)2-s + (−0.305 + 0.529i)4-s + (2.16 − 3.75i)5-s + 3.07i·8-s − 5.10i·10-s + (1.87 − 1.08i)11-s + (2.25 + 1.30i)13-s + (1.20 + 2.08i)16-s + 1.17·17-s − 2.41i·19-s + (1.32 + 2.29i)20-s + (1.27 − 2.20i)22-s + (3.16 + 1.82i)23-s + (−6.88 − 11.9i)25-s + 3.06·26-s + ⋯
L(s)  = 1  + (0.721 − 0.416i)2-s + (−0.152 + 0.264i)4-s + (0.968 − 1.67i)5-s + 1.08i·8-s − 1.61i·10-s + (0.564 − 0.325i)11-s + (0.624 + 0.360i)13-s + (0.300 + 0.520i)16-s + 0.284·17-s − 0.554i·19-s + (0.296 + 0.513i)20-s + (0.271 − 0.470i)22-s + (0.659 + 0.380i)23-s + (−1.37 − 2.38i)25-s + 0.601·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1323\)    =    \(3^{3} \cdot 7^{2}\)
Sign: $0.424 + 0.905i$
Analytic conductor: \(10.5642\)
Root analytic conductor: \(3.25026\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1323} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1323,\ (\ :1/2),\ 0.424 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.857977926\)
\(L(\frac12)\) \(\approx\) \(2.857977926\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.02 + 0.589i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-2.16 + 3.75i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.87 + 1.08i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.25 - 1.30i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.17T + 17T^{2} \)
19 \( 1 + 2.41iT - 19T^{2} \)
23 \( 1 + (-3.16 - 1.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.589 - 0.340i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.67 - 3.27i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.10T + 37T^{2} \)
41 \( 1 + (-3.68 + 6.38i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.12 + 3.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.57 + 6.18i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 3.23iT - 53T^{2} \)
59 \( 1 + (2.91 - 5.05i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.21 + 3.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.32 - 5.76i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.95iT - 71T^{2} \)
73 \( 1 - 11.9iT - 73T^{2} \)
79 \( 1 + (-4.87 - 8.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.796 + 1.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6.09T + 89T^{2} \)
97 \( 1 + (2.36 - 1.36i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.271947183545096186276679751181, −8.740905693844836476975877516069, −8.263818705881731525313144083500, −6.85922976787149606900249128270, −5.74434645025503483070071035220, −5.17412030920416849836653483960, −4.40865913942378587020226705522, −3.51268099307287506940627602610, −2.17101103456404037588021240616, −1.09531857698953364692024132954, 1.49903271678938458736534699407, 2.87252015407975084884392478206, 3.66932062058610970103641984521, 4.81382145904892204792748847426, 5.91295260907701609661948814594, 6.29057725369374203135890661853, 6.94073360679247336275895067738, 7.88751767236540388776696437614, 9.271607280249109970114830002878, 9.836848311598927271843775790779

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