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Properties

Label 2-968-88.5-c1-0-81
Degree $2$
Conductor $968$
Sign $0.388 + 0.921i$
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  + (−0.547 + 1.30i)2-s + (1.64 − 2.26i)3-s + (−1.39 − 1.42i)4-s + (−1.10 − 0.358i)5-s + (2.05 + 3.38i)6-s + (2.45 − 1.78i)7-s + (2.62 − 1.04i)8-s + (−1.49 − 4.59i)9-s + (1.07 − 1.24i)10-s + (−5.53 + 0.819i)12-s + (3.94 − 1.28i)13-s + (0.978 + 4.16i)14-s + (−2.62 + 1.90i)15-s + (−0.0808 + 3.99i)16-s + (−2.25 + 6.95i)17-s + (6.80 + 0.570i)18-s + ⋯
L(s)  = 1  + (−0.387 + 0.921i)2-s + (0.949 − 1.30i)3-s + (−0.699 − 0.714i)4-s + (−0.492 − 0.160i)5-s + (0.837 + 1.38i)6-s + (0.926 − 0.672i)7-s + (0.929 − 0.368i)8-s + (−0.497 − 1.53i)9-s + (0.338 − 0.392i)10-s + (−1.59 + 0.236i)12-s + (1.09 − 0.355i)13-s + (0.261 + 1.11i)14-s + (−0.677 + 0.492i)15-s + (−0.0202 + 0.999i)16-s + (−0.547 + 1.68i)17-s + (1.60 + 0.134i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.388 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.41562 - 0.938911i\)
\(L(\frac12)\) \(\approx\) \(1.41562 - 0.938911i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.547 - 1.30i)T \)
11 \( 1 \)
good3 \( 1 + (-1.64 + 2.26i)T + (-0.927 - 2.85i)T^{2} \)
5 \( 1 + (1.10 + 0.358i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.45 + 1.78i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-3.94 + 1.28i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (2.25 - 6.95i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.80 + 3.86i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 5.24T + 23T^{2} \)
29 \( 1 + (1.79 + 2.47i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.98 + 6.10i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.97 + 5.46i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.01 - 0.737i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 7.39iT - 43T^{2} \)
47 \( 1 + (0.670 + 0.486i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (7.52 - 2.44i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.00 + 4.13i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.97 + 1.61i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 - 2.79iT - 67T^{2} \)
71 \( 1 + (-2.34 + 7.21i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.01 + 0.737i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.51 - 13.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.852 - 0.277i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 - 0.656T + 89T^{2} \)
97 \( 1 + (-3.44 - 10.5i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.373785945884573109736131199406, −8.630717305280832832291292342328, −7.940384308363321142797703575307, −7.67949634544855522187158667884, −6.71481331556789557986998178338, −5.92894114913292902375772479708, −4.56227126039941829403760515028, −3.60116657312608866342175881283, −1.85669398243988790722726415953, −0.891128559317786626710670350955, 1.71863748490239200381283463056, 3.03336070389486050833716814773, 3.58044759780141858339493265531, 4.66051259631704584238895763176, 5.25797260893148672303268954624, 7.22975588235846180015828111564, 8.122216000198745352689049261585, 8.916343875948072932686919936152, 9.136512081797221064647611465306, 10.14053218187446526271665687997

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