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Properties

Label 2-704-11.4-c1-0-9
Degree $2$
Conductor $704$
Sign $0.220 - 0.975i$
Analytic cond. $5.62146$
Root an. cond. $2.37096$
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  + (−0.5 + 1.53i)3-s + (2.61 − 1.90i)5-s + (1.38 + 4.25i)7-s + (0.309 + 0.224i)9-s + (−3.30 + 0.224i)11-s + (1 + 0.726i)13-s + (1.61 + 4.97i)15-s + (−1.5 + 1.08i)17-s + (1.66 − 5.11i)19-s − 7.23·21-s + 5.23·23-s + (1.69 − 5.20i)25-s + (−4.42 + 3.21i)27-s + (2.61 + 8.05i)29-s + (−0.381 − 0.277i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.888i)3-s + (1.17 − 0.850i)5-s + (0.522 + 1.60i)7-s + (0.103 + 0.0748i)9-s + (−0.997 + 0.0676i)11-s + (0.277 + 0.201i)13-s + (0.417 + 1.28i)15-s + (−0.363 + 0.264i)17-s + (0.381 − 1.17i)19-s − 1.57·21-s + 1.09·23-s + (0.338 − 1.04i)25-s + (−0.851 + 0.619i)27-s + (0.486 + 1.49i)29-s + (−0.0686 − 0.0498i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $0.220 - 0.975i$
Analytic conductor: \(5.62146\)
Root analytic conductor: \(2.37096\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 704,\ (\ :1/2),\ 0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34802 + 1.07695i\)
\(L(\frac12)\) \(\approx\) \(1.34802 + 1.07695i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (3.30 - 0.224i)T \)
good3 \( 1 + (0.5 - 1.53i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-2.61 + 1.90i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.38 - 4.25i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1 - 0.726i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.5 - 1.08i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.66 + 5.11i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 5.23T + 23T^{2} \)
29 \( 1 + (-2.61 - 8.05i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.381 + 0.277i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.85 + 8.78i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.26 - 6.96i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 4.09T + 43T^{2} \)
47 \( 1 + (-2 + 6.15i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.23 - 3.80i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.80 - 8.64i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2 + 1.45i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 1.38T + 67T^{2} \)
71 \( 1 + (3 - 2.17i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.26 + 3.88i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.85 - 3.52i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.92 - 1.40i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 6.38T + 89T^{2} \)
97 \( 1 + (4.54 + 3.30i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.60349844642433598673145660249, −9.636527258309794792480982756712, −8.967097209800425767336308144332, −8.529531590266046994504209166925, −7.02171510610364273264067154243, −5.61406603786829643624574144319, −5.28125678972294138950556970003, −4.63969724399318743637758225639, −2.81353811862754280019985469476, −1.74067477481115171052631235867, 1.01852691685013715511785565957, 2.18413734609819484813611503590, 3.54715254853780312555648317694, 4.92530195875929471317692947020, 6.03070855415146616147687034428, 6.79192909273045539180253645544, 7.43879931128302334637090420754, 8.215896317841589645428950460850, 9.810286264312575674136963582199, 10.27155438338306189034454607295

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