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Properties

Label 2-966-21.20-c1-0-19
Degree $2$
Conductor $966$
Sign $0.897 + 0.440i$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
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  + i·2-s + (−1.37 − 1.05i)3-s − 4-s − 2.38·5-s + (1.05 − 1.37i)6-s + (−2.59 + 0.527i)7-s − i·8-s + (0.764 + 2.90i)9-s − 2.38i·10-s + 4.24i·11-s + (1.37 + 1.05i)12-s + 0.530i·13-s + (−0.527 − 2.59i)14-s + (3.27 + 2.52i)15-s + 16-s − 2.11·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.792 − 0.610i)3-s − 0.5·4-s − 1.06·5-s + (0.431 − 0.560i)6-s + (−0.979 + 0.199i)7-s − 0.353i·8-s + (0.254 + 0.966i)9-s − 0.754i·10-s + 1.28i·11-s + (0.396 + 0.305i)12-s + 0.147i·13-s + (−0.141 − 0.692i)14-s + (0.845 + 0.651i)15-s + 0.250·16-s − 0.512·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.897 + 0.440i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.897 + 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.461352 - 0.106954i\)
\(L(\frac12)\) \(\approx\) \(0.461352 - 0.106954i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (1.37 + 1.05i)T \)
7 \( 1 + (2.59 - 0.527i)T \)
23 \( 1 - iT \)
good5 \( 1 + 2.38T + 5T^{2} \)
11 \( 1 - 4.24iT - 11T^{2} \)
13 \( 1 - 0.530iT - 13T^{2} \)
17 \( 1 + 2.11T + 17T^{2} \)
19 \( 1 + 2.73iT - 19T^{2} \)
29 \( 1 + 4.13iT - 29T^{2} \)
31 \( 1 + 8.63iT - 31T^{2} \)
37 \( 1 - 1.99T + 37T^{2} \)
41 \( 1 - 8.92T + 41T^{2} \)
43 \( 1 - 8.94T + 43T^{2} \)
47 \( 1 + 5.75T + 47T^{2} \)
53 \( 1 + 2.38iT - 53T^{2} \)
59 \( 1 + 4.97T + 59T^{2} \)
61 \( 1 - 3.14iT - 61T^{2} \)
67 \( 1 + 7.15T + 67T^{2} \)
71 \( 1 + 11.1iT - 71T^{2} \)
73 \( 1 - 5.68iT - 73T^{2} \)
79 \( 1 - 10.5T + 79T^{2} \)
83 \( 1 + 1.92T + 83T^{2} \)
89 \( 1 - 17.8T + 89T^{2} \)
97 \( 1 - 9.07iT - 97T^{2} \)
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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

−9.806390363146128908531804473950, −9.130109295117636340755132350075, −7.78330862790430065862663407216, −7.47867538264448858978166556296, −6.57698916233695075966472274878, −5.92058790488666729063673792046, −4.69300572507160577799477121926, −4.02762994535554171535195356611, −2.38298362672689134189025876352, −0.37344150622309600804601073170, 0.797866506778339638009561802683, 3.09137157206852398246402820413, 3.69102865195109838068734448536, 4.54081377632512849495808275065, 5.70475826797281226527287944571, 6.49065319238461178267851007469, 7.61335584782663401457786230149, 8.703930061868199661223330579876, 9.341159159946542834271287165551, 10.40774463075123231188015489022

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