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Properties

Label 2-968-88.37-c1-0-47
Degree $2$
Conductor $968$
Sign $-0.257 - 0.966i$
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  + (1.05 + 0.940i)2-s + (2.42 + 0.789i)3-s + (0.229 + 1.98i)4-s + (−1.58 − 2.18i)5-s + (1.82 + 3.11i)6-s + (1.22 + 3.75i)7-s + (−1.62 + 2.31i)8-s + (2.85 + 2.07i)9-s + (0.379 − 3.79i)10-s + (−1.01 + 5.00i)12-s + (−0.257 + 0.354i)13-s + (−2.24 + 5.11i)14-s + (−2.12 − 6.54i)15-s + (−3.89 + 0.911i)16-s + (1.93 − 1.40i)17-s + (1.06 + 4.87i)18-s + ⋯
L(s)  = 1  + (0.746 + 0.665i)2-s + (1.40 + 0.455i)3-s + (0.114 + 0.993i)4-s + (−0.708 − 0.975i)5-s + (0.744 + 1.27i)6-s + (0.461 + 1.41i)7-s + (−0.575 + 0.817i)8-s + (0.951 + 0.691i)9-s + (0.119 − 1.19i)10-s + (−0.291 + 1.44i)12-s + (−0.0713 + 0.0982i)13-s + (−0.600 + 1.36i)14-s + (−0.549 − 1.69i)15-s + (−0.973 + 0.227i)16-s + (0.469 − 0.341i)17-s + (0.250 + 1.14i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.257 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.257 - 0.966i$
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: \((2,\ 968,\ (\ :1/2),\ -0.257 - 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09678 + 2.72737i\)
\(L(\frac12)\) \(\approx\) \(2.09678 + 2.72737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.05 - 0.940i)T \)
11 \( 1 \)
good3 \( 1 + (-2.42 - 0.789i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (1.58 + 2.18i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.22 - 3.75i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.257 - 0.354i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.93 + 1.40i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-5.08 - 1.65i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 1.35T + 23T^{2} \)
29 \( 1 + (0.140 - 0.0455i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.28 + 0.930i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (7.01 - 2.28i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.11 - 3.43i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.62iT - 43T^{2} \)
47 \( 1 + (-0.934 + 2.87i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.72 + 3.75i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-11.8 + 3.86i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.387 - 0.533i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 3.00iT - 67T^{2} \)
71 \( 1 + (-8.54 + 6.21i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-3.28 - 10.1i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.21 - 3.78i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (1.34 + 1.84i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + (7.02 + 5.10i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.813270289161437708657877940238, −9.051254534739185094642193336966, −8.363337493370557567992989514090, −8.117434350976789137811576719671, −7.08709194613617657502446969509, −5.55864795740778992609777031850, −5.04778916696268232736399902349, −3.99064068948784883135594421071, −3.20089878540191214950783928037, −2.12813940612235077665343608145, 1.19311592683313511285355194032, 2.48736077049163726945352131877, 3.52772706649687194058949850904, 3.79846344089675949583801561242, 5.11317985249513678348951672680, 6.63348388489791525576104389785, 7.40331263287987635292246047631, 7.78422379170676187478026270282, 9.004267269300669957557358870388, 9.995091517137937503272889482517

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