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Properties

Label 2-968-88.37-c1-0-27
Degree $2$
Conductor $968$
Sign $-0.114 + 0.993i$
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  + (−0.930 − 1.06i)2-s + (−1.78 − 0.580i)3-s + (−0.268 + 1.98i)4-s + (−0.289 − 0.398i)5-s + (1.04 + 2.44i)6-s + (0.0540 + 0.166i)7-s + (2.36 − 1.55i)8-s + (0.432 + 0.314i)9-s + (−0.155 + 0.679i)10-s + (1.63 − 3.38i)12-s + (−3.24 + 4.46i)13-s + (0.126 − 0.212i)14-s + (0.286 + 0.881i)15-s + (−3.85 − 1.06i)16-s + (0.548 − 0.398i)17-s + (−0.0677 − 0.752i)18-s + ⋯
L(s)  = 1  + (−0.657 − 0.753i)2-s + (−1.03 − 0.335i)3-s + (−0.134 + 0.990i)4-s + (−0.129 − 0.178i)5-s + (0.426 + 0.998i)6-s + (0.0204 + 0.0628i)7-s + (0.834 − 0.550i)8-s + (0.144 + 0.104i)9-s + (−0.0490 + 0.214i)10-s + (0.470 − 0.977i)12-s + (−0.899 + 1.23i)13-s + (0.0338 − 0.0567i)14-s + (0.0739 + 0.227i)15-s + (−0.963 − 0.265i)16-s + (0.133 − 0.0967i)17-s + (−0.0159 − 0.177i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.114 + 0.993i$
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: \((2,\ 968,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.381085 - 0.427354i\)
\(L(\frac12)\) \(\approx\) \(0.381085 - 0.427354i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.930 + 1.06i)T \)
11 \( 1 \)
good3 \( 1 + (1.78 + 0.580i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.289 + 0.398i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.0540 - 0.166i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.24 - 4.46i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.548 + 0.398i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.415 - 0.135i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 6.88T + 23T^{2} \)
29 \( 1 + (1.34 - 0.436i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.37 - 2.45i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (6.20 - 2.01i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.02 - 9.30i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 8.22iT - 43T^{2} \)
47 \( 1 + (-2.39 + 7.37i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.83 + 8.02i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-8.99 + 2.92i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.64 + 7.77i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.24iT - 67T^{2} \)
71 \( 1 + (3.73 - 2.71i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.11 + 6.51i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.48 - 3.98i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.28 + 5.89i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 9.61T + 89T^{2} \)
97 \( 1 + (2.71 + 1.97i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.940138833660158862785940045346, −9.008588372333569104788972730849, −8.369718867818418336556641244942, −6.99700372323269766432499088967, −6.80779873517000523849220827538, −5.28067518932488304100665425056, −4.53054179596845369874895802104, −3.20950152647845589524779521733, −1.90913111210657682447045915628, −0.54730189713730599744950047830, 0.864829168279561076834051838048, 2.78227518717030841107620136272, 4.46491185766010063740006112670, 5.38483295591697987789083098039, 5.76976626772852761338318835281, 7.00176862349982591108561500852, 7.52601245092221989690214784387, 8.561771476084473382233676355294, 9.432865754737565360439393980312, 10.39221608572303312175605483772

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