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Properties

Label 2-968-88.37-c1-0-16
Degree $2$
Conductor $968$
Sign $-0.990 - 0.135i$
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  + (−1.07 + 0.922i)2-s + (0.513 + 0.167i)3-s + (0.296 − 1.97i)4-s + (1.70 + 2.34i)5-s + (−0.704 + 0.295i)6-s + (0.804 + 2.47i)7-s + (1.50 + 2.39i)8-s + (−2.19 − 1.59i)9-s + (−3.99 − 0.941i)10-s + (0.482 − 0.967i)12-s + (−1.53 + 2.11i)13-s + (−3.14 − 1.91i)14-s + (0.484 + 1.49i)15-s + (−3.82 − 1.17i)16-s + (−5.57 + 4.05i)17-s + (3.81 − 0.316i)18-s + ⋯
L(s)  = 1  + (−0.757 + 0.652i)2-s + (0.296 + 0.0964i)3-s + (0.148 − 0.988i)4-s + (0.762 + 1.04i)5-s + (−0.287 + 0.120i)6-s + (0.304 + 0.936i)7-s + (0.532 + 0.846i)8-s + (−0.730 − 0.530i)9-s + (−1.26 − 0.297i)10-s + (0.139 − 0.279i)12-s + (−0.426 + 0.587i)13-s + (−0.841 − 0.510i)14-s + (0.125 + 0.384i)15-s + (−0.955 − 0.293i)16-s + (−1.35 + 0.982i)17-s + (0.899 − 0.0745i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $-0.990 - 0.135i$
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: \((2,\ 968,\ (\ :1/2),\ -0.990 - 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0614355 + 0.903460i\)
\(L(\frac12)\) \(\approx\) \(0.0614355 + 0.903460i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 - 0.922i)T \)
11 \( 1 \)
good3 \( 1 + (-0.513 - 0.167i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.70 - 2.34i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.804 - 2.47i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.53 - 2.11i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.57 - 4.05i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.52 + 0.494i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 4.01T + 23T^{2} \)
29 \( 1 + (1.89 - 0.617i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-4.19 - 3.04i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-8.16 + 2.65i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.15 + 3.56i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.47iT - 43T^{2} \)
47 \( 1 + (3.66 - 11.2i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.760 + 1.04i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.334 - 0.108i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.81 + 2.49i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + (2.80 - 2.03i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.412 - 1.26i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.49 + 2.54i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.39 - 12.9i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + (6.96 + 5.06i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.20624940136443477321547394959, −9.403471078120022672173060391174, −8.798260315804486363528076604905, −8.097045876194433797814837875509, −6.89843158062569062470154010222, −6.22504364283994691217410162772, −5.71933036531211814445239247082, −4.34433532004536671803550340007, −2.63400978736484874085094412454, −2.02504657424863396017107447802, 0.48155585998907098795985006452, 1.85523897334383829303268319913, 2.76376958078838868375225174443, 4.24458427174395705499797664271, 4.99916040076039946072057842547, 6.28692662262801769945188963058, 7.50008859539991880776423445573, 8.117356973220007918760913328384, 8.875987478116021690572688153420, 9.597810663440353236832365506933

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