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Properties

Label 2-968-968.395-c1-0-15
Degree $2$
Conductor $968$
Sign $-0.345 - 0.938i$
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  + (−1.39 − 0.201i)2-s − 1.12·3-s + (1.91 + 0.563i)4-s + (1.57 + 0.225i)6-s + (−2.57 − 1.17i)8-s − 1.74·9-s + (0.0401 − 3.31i)11-s + (−2.15 − 0.632i)12-s + (3.36 + 2.16i)16-s + (2.34 + 2.03i)17-s + (2.43 + 0.350i)18-s + (−1.28 + 1.11i)19-s + (−0.723 + 4.63i)22-s + (2.88 + 1.31i)24-s + (−3.27 + 3.77i)25-s + ⋯
L(s)  = 1  + (−0.989 − 0.142i)2-s − 0.647·3-s + (0.959 + 0.281i)4-s + (0.641 + 0.0922i)6-s + (−0.909 − 0.415i)8-s − 0.580·9-s + (0.0121 − 0.999i)11-s + (−0.621 − 0.182i)12-s + (0.841 + 0.540i)16-s + (0.569 + 0.493i)17-s + (0.574 + 0.0825i)18-s + (−0.295 + 0.256i)19-s + (−0.154 + 0.988i)22-s + (0.589 + 0.269i)24-s + (−0.654 + 0.755i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.345 - 0.938i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (395, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ -0.345 - 0.938i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.181541 + 0.260147i\)
\(L(\frac12)\) \(\approx\) \(0.181541 + 0.260147i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.201i)T \)
11 \( 1 + (-0.0401 + 3.31i)T \)
good3 \( 1 + 1.12T + 3T^{2} \)
5 \( 1 + (3.27 - 3.77i)T^{2} \)
7 \( 1 + (2.90 - 6.36i)T^{2} \)
13 \( 1 + (10.9 + 7.02i)T^{2} \)
17 \( 1 + (-2.34 - 2.03i)T + (2.41 + 16.8i)T^{2} \)
19 \( 1 + (1.28 - 1.11i)T + (2.70 - 18.8i)T^{2} \)
23 \( 1 + (-9.55 - 20.9i)T^{2} \)
29 \( 1 + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-26.0 + 16.7i)T^{2} \)
37 \( 1 + (-31.1 + 20.0i)T^{2} \)
41 \( 1 + (6.20 + 0.892i)T + (39.3 + 11.5i)T^{2} \)
43 \( 1 + (-1.82 - 0.831i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + (45.0 - 13.2i)T^{2} \)
53 \( 1 + (-22.0 + 48.2i)T^{2} \)
59 \( 1 + (0.252 + 1.75i)T + (-56.6 + 16.6i)T^{2} \)
61 \( 1 + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (1.99 - 13.8i)T + (-64.2 - 18.8i)T^{2} \)
71 \( 1 + (10.1 - 70.2i)T^{2} \)
73 \( 1 + (9.10 - 14.1i)T + (-30.3 - 66.4i)T^{2} \)
79 \( 1 + (-51.7 + 59.7i)T^{2} \)
83 \( 1 + (-1.52 - 2.37i)T + (-34.4 + 75.4i)T^{2} \)
89 \( 1 + (11.9 - 13.7i)T + (-12.6 - 88.0i)T^{2} \)
97 \( 1 + (7.56 - 16.5i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.35298557348264715663037002426, −9.460160189147064913204144084139, −8.547385671887674352262969357681, −8.014277275808677285918248777936, −6.91799254613789417819986763180, −6.01396611033909521149922959290, −5.46308860561340553117206005750, −3.77891581153013231129154406734, −2.75676074258336254285568451003, −1.23990400008949776986984883556, 0.23589720413242640797001595726, 1.86472272885354900938741200670, 3.07125310384131902043057344925, 4.68208028788443346993592707808, 5.64515326084799686021308606841, 6.46198097958184709682332802382, 7.26818887971479050178495421855, 8.120531507739621008548970129930, 8.959458377550130318131989049845, 9.849210151057983141130233716849

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