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Properties

Label 2-968-8.5-c1-0-74
Degree $2$
Conductor $968$
Sign $0.781 + 0.623i$
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  + (1.37 + 0.314i)2-s − 1.87i·3-s + (1.80 + 0.867i)4-s + 0.493i·5-s + (0.591 − 2.59i)6-s + 0.174·7-s + (2.21 + 1.76i)8-s − 0.534·9-s + (−0.155 + 0.679i)10-s + (1.63 − 3.38i)12-s − 5.51i·13-s + (0.240 + 0.0549i)14-s + 0.927·15-s + (2.49 + 3.12i)16-s − 0.678·17-s + (−0.736 − 0.168i)18-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s − 1.08i·3-s + (0.900 + 0.433i)4-s + 0.220i·5-s + (0.241 − 1.05i)6-s + 0.0660·7-s + (0.781 + 0.623i)8-s − 0.178·9-s + (−0.0490 + 0.214i)10-s + (0.470 − 0.977i)12-s − 1.53i·13-s + (0.0644 + 0.0146i)14-s + 0.239·15-s + (0.623 + 0.781i)16-s − 0.164·17-s + (−0.173 − 0.0396i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 968 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(968\)    =    \(2^{3} \cdot 11^{2}\)
Sign: $0.781 + 0.623i$
Analytic conductor: \(7.72951\)
Root analytic conductor: \(2.78020\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{968} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 968,\ (\ :1/2),\ 0.781 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.96202 - 1.03633i\)
\(L(\frac12)\) \(\approx\) \(2.96202 - 1.03633i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 - 0.314i)T \)
11 \( 1 \)
good3 \( 1 + 1.87iT - 3T^{2} \)
5 \( 1 - 0.493iT - 5T^{2} \)
7 \( 1 - 0.174T + 7T^{2} \)
13 \( 1 + 5.51iT - 13T^{2} \)
17 \( 1 + 0.678T + 17T^{2} \)
19 \( 1 - 0.437iT - 19T^{2} \)
23 \( 1 - 6.88T + 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 + 4.17T + 31T^{2} \)
37 \( 1 - 6.52iT - 37T^{2} \)
41 \( 1 + 9.78T + 41T^{2} \)
43 \( 1 + 8.22iT - 43T^{2} \)
47 \( 1 - 7.75T + 47T^{2} \)
53 \( 1 - 9.91iT - 53T^{2} \)
59 \( 1 + 9.45iT - 59T^{2} \)
61 \( 1 - 9.60iT - 61T^{2} \)
67 \( 1 + 5.24iT - 67T^{2} \)
71 \( 1 - 4.61T + 71T^{2} \)
73 \( 1 + 6.85T + 73T^{2} \)
79 \( 1 + 6.77T + 79T^{2} \)
83 \( 1 - 7.28iT - 83T^{2} \)
89 \( 1 + 9.61T + 89T^{2} \)
97 \( 1 - 3.35T + 97T^{2} \)
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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

−10.28071517965207541241693706936, −8.758574808239847694797530786248, −7.919270338354706520092549661030, −7.17222617322004455602549030534, −6.63464133362946486754019910229, −5.61076259342448570368284218468, −4.84147933377345988747747312986, −3.43692019599691049251092601791, −2.60464750423640798471752386560, −1.26001895355764569819536841474, 1.64239094254195054810400078110, 3.04143979717963280657257944873, 4.04771529307581140014110776940, 4.68611493365299725731799295830, 5.38185629742278033600226171642, 6.63047710731879886484650932140, 7.23836636624702271554847000713, 8.749424306163128444719080453125, 9.412114372747369386056249802151, 10.21965636984578522824572563519

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