Properties

Label 2-688-43.42-c4-0-25
Degree $2$
Conductor $688$
Sign $-0.285 - 0.958i$
Analytic cond. $71.1185$
Root an. cond. $8.43318$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.93i·3-s + 22.3i·5-s − 51.9i·7-s + 32.8·9-s + 25.0·11-s − 38.7·13-s − 155.·15-s + 111.·17-s + 238. i·19-s + 360.·21-s − 823.·23-s + 125.·25-s + 790. i·27-s − 424. i·29-s + 1.44e3·31-s + ⋯
L(s)  = 1  + 0.771i·3-s + 0.894i·5-s − 1.06i·7-s + 0.405·9-s + 0.206·11-s − 0.229·13-s − 0.689·15-s + 0.385·17-s + 0.661i·19-s + 0.818·21-s − 1.55·23-s + 0.200·25-s + 1.08i·27-s − 0.505i·29-s + 1.50·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(688\)    =    \(2^{4} \cdot 43\)
Sign: $-0.285 - 0.958i$
Analytic conductor: \(71.1185\)
Root analytic conductor: \(8.43318\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{688} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 688,\ (\ :2),\ -0.285 - 0.958i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.030959712\)
\(L(\frac12)\) \(\approx\) \(2.030959712\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 + (-528. - 1.77e3i)T \)
good3 \( 1 - 6.93iT - 81T^{2} \)
5 \( 1 - 22.3iT - 625T^{2} \)
7 \( 1 + 51.9iT - 2.40e3T^{2} \)
11 \( 1 - 25.0T + 1.46e4T^{2} \)
13 \( 1 + 38.7T + 2.85e4T^{2} \)
17 \( 1 - 111.T + 8.35e4T^{2} \)
19 \( 1 - 238. iT - 1.30e5T^{2} \)
23 \( 1 + 823.T + 2.79e5T^{2} \)
29 \( 1 + 424. iT - 7.07e5T^{2} \)
31 \( 1 - 1.44e3T + 9.23e5T^{2} \)
37 \( 1 + 626. iT - 1.87e6T^{2} \)
41 \( 1 - 580.T + 2.82e6T^{2} \)
47 \( 1 - 170.T + 4.87e6T^{2} \)
53 \( 1 - 4.30e3T + 7.89e6T^{2} \)
59 \( 1 - 65.4T + 1.21e7T^{2} \)
61 \( 1 - 3.89e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.44e3T + 2.01e7T^{2} \)
71 \( 1 - 5.84e3iT - 2.54e7T^{2} \)
73 \( 1 - 4.77e3iT - 2.83e7T^{2} \)
79 \( 1 + 1.01e4T + 3.89e7T^{2} \)
83 \( 1 + 4.43e3T + 4.74e7T^{2} \)
89 \( 1 - 9.87e3iT - 6.27e7T^{2} \)
97 \( 1 + 7.76e3T + 8.85e7T^{2} \)
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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

−10.06558258556716760786859828733, −9.779147816169240741438355681886, −8.339558637717355319911597607659, −7.43884035962960671978328113140, −6.72006303448742345702528701513, −5.66822087071411204814866944855, −4.27462236965125938767784520756, −3.88546374148794022366628746515, −2.62750910840383320622646273003, −1.08810476257767881809872177691, 0.55170700119465375561206059233, 1.64391778317133922858775105930, 2.63182112105232419252219778883, 4.18538665315181204424325584689, 5.15792989837871894088465808253, 6.07076425373276256598991634219, 6.97808386844554765653218393815, 8.019154905847007593527673563879, 8.652839027495038410398006861380, 9.503893507376920667160804405401

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