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Properties

Label 2-968-88.53-c1-0-48
Degree $2$
Conductor $968$
Sign $0.350 + 0.936i$
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  + (−1.40 − 0.116i)2-s + (0.317 + 0.437i)3-s + (1.97 + 0.329i)4-s + (−2.75 + 0.896i)5-s + (−0.396 − 0.653i)6-s + (2.10 + 1.53i)7-s + (−2.74 − 0.694i)8-s + (0.836 − 2.57i)9-s + (3.99 − 0.941i)10-s + (0.482 + 0.967i)12-s + (−2.48 − 0.808i)13-s + (−2.79 − 2.40i)14-s + (−1.26 − 0.921i)15-s + (3.78 + 1.29i)16-s + (−2.12 − 6.55i)17-s + (−1.47 + 3.53i)18-s + ⋯
L(s)  = 1  + (−0.996 − 0.0825i)2-s + (0.183 + 0.252i)3-s + (0.986 + 0.164i)4-s + (−1.23 + 0.400i)5-s + (−0.161 − 0.266i)6-s + (0.796 + 0.578i)7-s + (−0.969 − 0.245i)8-s + (0.278 − 0.858i)9-s + (1.26 − 0.297i)10-s + (0.139 + 0.279i)12-s + (−0.690 − 0.224i)13-s + (−0.745 − 0.642i)14-s + (−0.327 − 0.237i)15-s + (0.945 + 0.324i)16-s + (−0.516 − 1.58i)17-s + (−0.348 + 0.832i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.350 + 0.936i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (493, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.350 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.537402 - 0.372555i\)
\(L(\frac12)\) \(\approx\) \(0.537402 - 0.372555i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.116i)T \)
11 \( 1 \)
good3 \( 1 + (-0.317 - 0.437i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (2.75 - 0.896i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.10 - 1.53i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (2.48 + 0.808i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.12 + 6.55i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-0.940 - 1.29i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.01T + 23T^{2} \)
29 \( 1 + (-1.17 + 1.61i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.60 - 4.92i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-5.04 + 6.94i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-3.03 + 2.20i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 2.47iT - 43T^{2} \)
47 \( 1 + (-9.58 + 6.96i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (1.23 + 0.399i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.206 - 0.284i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.93 - 0.952i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 + (-1.07 - 3.30i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.07 - 0.784i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.33 - 4.11i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-15.2 + 4.93i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + (-2.66 + 8.19i)T + (-78.4 - 57.0i)T^{2} \)
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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

−9.692381471622904967282834047841, −9.043677701869871690208817387635, −8.221180002948497673935971348572, −7.44898524282921817169754342841, −6.92301394031538545814148173818, −5.61431875495108835501644014222, −4.34883446565555660037405090749, −3.30542202539172204085084438984, −2.26923989287013007862231973617, −0.45057762502707515704530757393, 1.24047084859416047104056836648, 2.39711412853311661854925024833, 4.00569318620182944665489273064, 4.75513414532955054538399318579, 6.16369923812929263523257274816, 7.28312529379636350099906040722, 7.945335669677209579256779884739, 8.117448765731390557329146033676, 9.181677603712850512198474104555, 10.26433158548108358046937901775

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