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Properties

Label 2-693-11.4-c1-0-18
Degree $2$
Conductor $693$
Sign $0.855 + 0.517i$
Analytic cond. $5.53363$
Root an. cond. $2.35236$
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  + (0.727 + 0.528i)2-s + (−0.367 − 1.13i)4-s + (−0.650 + 0.472i)5-s + (0.309 + 0.951i)7-s + (0.887 − 2.73i)8-s − 0.723·10-s + (3.31 − 0.199i)11-s + (−2.38 − 1.73i)13-s + (−0.278 + 0.855i)14-s + (0.163 − 0.119i)16-s + (2.67 − 1.94i)17-s + (2.45 − 7.55i)19-s + (0.774 + 0.562i)20-s + (2.51 + 1.60i)22-s + 7.53·23-s + ⋯
L(s)  = 1  + (0.514 + 0.373i)2-s + (−0.183 − 0.566i)4-s + (−0.290 + 0.211i)5-s + (0.116 + 0.359i)7-s + (0.313 − 0.965i)8-s − 0.228·10-s + (0.998 − 0.0602i)11-s + (−0.661 − 0.480i)13-s + (−0.0743 + 0.228i)14-s + (0.0409 − 0.0297i)16-s + (0.648 − 0.471i)17-s + (0.562 − 1.73i)19-s + (0.173 + 0.125i)20-s + (0.536 + 0.342i)22-s + 1.57·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.855 + 0.517i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (631, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.855 + 0.517i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78602 - 0.498386i\)
\(L(\frac12)\) \(\approx\) \(1.78602 - 0.498386i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-3.31 + 0.199i)T \)
good2 \( 1 + (-0.727 - 0.528i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (0.650 - 0.472i)T + (1.54 - 4.75i)T^{2} \)
13 \( 1 + (2.38 + 1.73i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.67 + 1.94i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.45 + 7.55i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 7.53T + 23T^{2} \)
29 \( 1 + (-0.0724 - 0.222i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.22 + 2.34i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.165 + 0.510i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.47 + 4.55i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 3.16T + 43T^{2} \)
47 \( 1 + (-3.40 + 10.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-9.49 - 6.89i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.62 - 4.99i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.95 - 2.14i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + (8.36 - 6.07i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.75 - 8.47i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.77 - 4.92i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (5.87 - 4.26i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 8.03T + 89T^{2} \)
97 \( 1 + (15.5 + 11.3i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.41245754010964280867666883026, −9.323749963960820304756922451709, −8.972738131079505653274846114432, −7.25263471520008240074775219751, −7.05986374624694207515096728701, −5.66510325426693209725910006211, −5.11234475406836017977863477493, −4.01935965753391422523422431833, −2.80418003890921067950380306389, −0.958272979930099444101306062250, 1.55577959595657241991024621669, 3.13267789185970017695042853793, 3.99984453197988356992405666854, 4.75523258528294778448651515156, 5.93064218952551963115125999336, 7.19665025316318175919375091063, 7.88301671410741975743037455931, 8.795128691527715106411180225340, 9.677303971597249141238021452568, 10.70368745567849841276980274491

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