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Properties

Label 2-966-21.20-c1-0-30
Degree $2$
Conductor $966$
Sign $0.937 - 0.347i$
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 + (0.740 + 1.56i)3-s − 4-s + 3.67·5-s + (1.56 − 0.740i)6-s + (−0.229 + 2.63i)7-s + i·8-s + (−1.90 + 2.31i)9-s − 3.67i·10-s − 3.25i·11-s + (−0.740 − 1.56i)12-s − 0.990i·13-s + (2.63 + 0.229i)14-s + (2.72 + 5.75i)15-s + 16-s + 6.53·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.427 + 0.903i)3-s − 0.5·4-s + 1.64·5-s + (0.639 − 0.302i)6-s + (−0.0866 + 0.996i)7-s + 0.353i·8-s + (−0.634 + 0.773i)9-s − 1.16i·10-s − 0.981i·11-s + (−0.213 − 0.451i)12-s − 0.274i·13-s + (0.704 + 0.0613i)14-s + (0.703 + 1.48i)15-s + 0.250·16-s + 1.58·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.937 - 0.347i$
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: \((2,\ 966,\ (\ :1/2),\ 0.937 - 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.26407 + 0.406376i\)
\(L(\frac12)\) \(\approx\) \(2.26407 + 0.406376i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.740 - 1.56i)T \)
7 \( 1 + (0.229 - 2.63i)T \)
23 \( 1 - iT \)
good5 \( 1 - 3.67T + 5T^{2} \)
11 \( 1 + 3.25iT - 11T^{2} \)
13 \( 1 + 0.990iT - 13T^{2} \)
17 \( 1 - 6.53T + 17T^{2} \)
19 \( 1 + 1.80iT - 19T^{2} \)
29 \( 1 - 1.83iT - 29T^{2} \)
31 \( 1 - 7.84iT - 31T^{2} \)
37 \( 1 + 0.190T + 37T^{2} \)
41 \( 1 + 2.12T + 41T^{2} \)
43 \( 1 - 6.53T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 - 7.60iT - 53T^{2} \)
59 \( 1 + 4.25T + 59T^{2} \)
61 \( 1 + 3.34iT - 61T^{2} \)
67 \( 1 + 3.26T + 67T^{2} \)
71 \( 1 + 2.80iT - 71T^{2} \)
73 \( 1 + 0.439iT - 73T^{2} \)
79 \( 1 + 5.24T + 79T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 - 8.22T + 89T^{2} \)
97 \( 1 + 12.5iT - 97T^{2} \)
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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

−10.09910674771408915699968326127, −9.260578874234713397852726450073, −8.908516409890913986557989234148, −7.957960603299782384671021346098, −6.22270089080206218901123166384, −5.50427996628609863184325152396, −4.98724558003332716418697083808, −3.29874812089598448118128214947, −2.81804472210490767293416685525, −1.59417758925865815101144858806, 1.18056297219099000180493325881, 2.22138969433060596344227095880, 3.62427721537810562769652009297, 4.96872328942397889163129284961, 5.98011910151162299237205010024, 6.53558408744627230920365086188, 7.44370432582828539416088339573, 7.996678797226808837630776995806, 9.250741735680610877032185597036, 9.779773491170527874975695403860

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