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Properties

Label 2-693-11.5-c1-0-0
Degree $2$
Conductor $693$
Sign $0.271 - 0.962i$
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  + (−0.557 − 1.71i)2-s + (−1.01 + 0.740i)4-s + (−0.858 + 2.64i)5-s + (0.809 − 0.587i)7-s + (−1.08 − 0.785i)8-s + 5.01·10-s + (−3.22 − 0.754i)11-s + (0.714 + 2.19i)13-s + (−1.46 − 1.06i)14-s + (−1.52 + 4.69i)16-s + (−0.822 + 2.53i)17-s + (−6.53 − 4.75i)19-s + (−1.08 − 3.32i)20-s + (0.507 + 5.96i)22-s + 2.43·23-s + ⋯
L(s)  = 1  + (−0.394 − 1.21i)2-s + (−0.509 + 0.370i)4-s + (−0.383 + 1.18i)5-s + (0.305 − 0.222i)7-s + (−0.382 − 0.277i)8-s + 1.58·10-s + (−0.973 − 0.227i)11-s + (0.198 + 0.610i)13-s + (−0.390 − 0.283i)14-s + (−0.381 + 1.17i)16-s + (−0.199 + 0.613i)17-s + (−1.50 − 1.08i)19-s + (−0.241 − 0.743i)20-s + (0.108 + 1.27i)22-s + 0.506·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.271 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.271 - 0.962i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.271 - 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.301582 + 0.228235i\)
\(L(\frac12)\) \(\approx\) \(0.301582 + 0.228235i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (3.22 + 0.754i)T \)
good2 \( 1 + (0.557 + 1.71i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.858 - 2.64i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (-0.714 - 2.19i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.822 - 2.53i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.53 + 4.75i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 - 2.43T + 23T^{2} \)
29 \( 1 + (6.11 - 4.44i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.85 - 8.79i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (9.05 - 6.58i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.242 + 0.176i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 7.29T + 43T^{2} \)
47 \( 1 + (-0.370 - 0.269i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.60 - 4.93i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.93 + 7.21i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.561 - 1.72i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.42T + 67T^{2} \)
71 \( 1 + (0.172 - 0.530i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.81 + 5.67i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.18 + 9.81i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.62 + 8.09i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 1.56T + 89T^{2} \)
97 \( 1 + (0.164 + 0.507i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.68371521160519534998707579058, −10.31428155780151178156325408641, −8.992558744569260777245508928555, −8.353503012700728901560607697969, −7.00863199105954735263105270389, −6.52096386749192394050659778753, −4.94607801641750854797996809909, −3.66377316211242155441888601482, −2.86957102061338496539719353512, −1.79192393997869248382494822952, 0.21183492081880816675925048667, 2.29406128211786242886843060870, 4.05510566888848348491487504172, 5.19926251900411055577974158145, 5.70010488541420613652822847686, 6.90474911944269619336728990889, 7.972448455139978694730412664419, 8.229132660758469528513611884089, 9.040367574914617387496700328061, 10.01349274193997871828639252548

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