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Properties

Label 2-968-88.37-c1-0-58
Degree $2$
Conductor $968$
Sign $0.805 + 0.592i$
Analytic cond. $7.72951$
Root an. cond. $2.78020$
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  + (−0.323 − 1.37i)2-s + (2.66 + 0.864i)3-s + (−1.79 + 0.889i)4-s + (0.681 + 0.937i)5-s + (0.330 − 3.94i)6-s + (−0.935 − 2.88i)7-s + (1.80 + 2.17i)8-s + (3.90 + 2.83i)9-s + (1.07 − 1.24i)10-s + (−5.53 + 0.819i)12-s + (−2.43 + 3.35i)13-s + (−3.66 + 2.21i)14-s + (1.00 + 3.08i)15-s + (2.41 − 3.18i)16-s + (5.91 − 4.29i)17-s + (2.64 − 6.29i)18-s + ⋯
L(s)  = 1  + (−0.228 − 0.973i)2-s + (1.53 + 0.499i)3-s + (−0.895 + 0.444i)4-s + (0.304 + 0.419i)5-s + (0.134 − 1.60i)6-s + (−0.353 − 1.08i)7-s + (0.637 + 0.770i)8-s + (1.30 + 0.946i)9-s + (0.338 − 0.392i)10-s + (−1.59 + 0.236i)12-s + (−0.675 + 0.929i)13-s + (−0.978 + 0.593i)14-s + (0.258 + 0.796i)15-s + (0.604 − 0.796i)16-s + (1.43 − 1.04i)17-s + (0.623 − 1.48i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21352 - 0.726751i\)
\(L(\frac12)\) \(\approx\) \(2.21352 - 0.726751i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.323 + 1.37i)T \)
11 \( 1 \)
good3 \( 1 + (-2.66 - 0.864i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-0.681 - 0.937i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.935 + 2.88i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.43 - 3.35i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.91 + 4.29i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.54 - 1.47i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 5.24T + 23T^{2} \)
29 \( 1 + (2.91 - 0.946i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-5.18 - 3.77i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (6.42 - 2.08i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.387 - 1.19i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 7.39iT - 43T^{2} \)
47 \( 1 + (-0.255 + 0.787i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.65 + 6.40i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (4.86 - 1.58i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.07 - 4.22i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 2.79iT - 67T^{2} \)
71 \( 1 + (6.13 - 4.45i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.387 + 1.19i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.8 + 8.59i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.527 + 0.725i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 0.656T + 89T^{2} \)
97 \( 1 + (9.01 + 6.54i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.01407084202130169705988601696, −9.367679525776764072637299828910, −8.546743465298145210993747414366, −7.54098164970426555124527955089, −7.04745987466417685990969775237, −5.10532092077425136018819892583, −4.19642509602627594123766418921, −3.21861815718874803595871097633, −2.79413989270423337905230887385, −1.33384289526855986679989681148, 1.30691504543707978161348389712, 2.73116196103406538647893775533, 3.62333346132972021387140932902, 5.23537329205464357174966030904, 5.73170805270793964743926051110, 7.03886363694102256937046740687, 7.72234864813951979506387215027, 8.370088773784968473554498302299, 9.137921523848863605648280838368, 9.531058177965738075200560588565

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