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Properties

Label 2-966-69.68-c1-0-38
Degree $2$
Conductor $966$
Sign $-0.693 + 0.720i$
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 + (−1.01 + 1.40i)3-s − 4-s + 1.51·5-s + (1.40 + 1.01i)6-s − i·7-s + i·8-s + (−0.945 − 2.84i)9-s − 1.51i·10-s − 1.20·11-s + (1.01 − 1.40i)12-s − 2.45·13-s − 14-s + (−1.53 + 2.12i)15-s + 16-s − 3.78·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.585 + 0.810i)3-s − 0.5·4-s + 0.675·5-s + (0.573 + 0.413i)6-s − 0.377i·7-s + 0.353i·8-s + (−0.315 − 0.949i)9-s − 0.477i·10-s − 0.362·11-s + (0.292 − 0.405i)12-s − 0.681·13-s − 0.267·14-s + (−0.395 + 0.547i)15-s + 0.250·16-s − 0.919·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.693 + 0.720i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.693 + 0.720i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282383 - 0.663601i\)
\(L(\frac12)\) \(\approx\) \(0.282383 - 0.663601i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.01 - 1.40i)T \)
7 \( 1 + iT \)
23 \( 1 + (-4.74 - 0.674i)T \)
good5 \( 1 - 1.51T + 5T^{2} \)
11 \( 1 + 1.20T + 11T^{2} \)
13 \( 1 + 2.45T + 13T^{2} \)
17 \( 1 + 3.78T + 17T^{2} \)
19 \( 1 + 4.80iT - 19T^{2} \)
29 \( 1 + 9.64iT - 29T^{2} \)
31 \( 1 + 0.842T + 31T^{2} \)
37 \( 1 - 9.11iT - 37T^{2} \)
41 \( 1 + 8.42iT - 41T^{2} \)
43 \( 1 - 6.02iT - 43T^{2} \)
47 \( 1 + 12.1iT - 47T^{2} \)
53 \( 1 + 9.05T + 53T^{2} \)
59 \( 1 + 8.50iT - 59T^{2} \)
61 \( 1 + 5.57iT - 61T^{2} \)
67 \( 1 + 13.3iT - 67T^{2} \)
71 \( 1 + 4.48iT - 71T^{2} \)
73 \( 1 + 8.07T + 73T^{2} \)
79 \( 1 - 9.34iT - 79T^{2} \)
83 \( 1 + 8.00T + 83T^{2} \)
89 \( 1 - 0.364T + 89T^{2} \)
97 \( 1 + 8.40iT - 97T^{2} \)
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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

−9.744828156798299624435228686553, −9.359949788121537231720579390575, −8.313234372319142602360009185819, −7.01597036931302277923457710933, −6.10468094730830285897632220499, −5.02775638150569999970704930561, −4.51646738873829049255382546305, −3.28280388862457147594998642895, −2.16230068058067401639794229949, −0.35223811499377454722370583394, 1.56934123700642020901122674460, 2.77869169353388061353494001965, 4.53401837728004851327243428739, 5.46741254778494871967974906071, 5.99084389931040893276767823717, 6.95197332689780536956447419656, 7.56668148999731654965666413141, 8.565428264336738342983014673446, 9.328266086780308316281046803821, 10.34158146041270986812107328183

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