Properties

Label 2-966-161.160-c1-0-24
Degree $2$
Conductor $966$
Sign $0.773 + 0.633i$
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  + 2-s i·3-s + 4-s + 2.93·5-s i·6-s + (0.912 − 2.48i)7-s + 8-s − 9-s + 2.93·10-s + 4.68i·11-s i·12-s − 0.902i·13-s + (0.912 − 2.48i)14-s − 2.93i·15-s + 16-s + 1.82·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577i·3-s + 0.5·4-s + 1.31·5-s − 0.408i·6-s + (0.344 − 0.938i)7-s + 0.353·8-s − 0.333·9-s + 0.927·10-s + 1.41i·11-s − 0.288i·12-s − 0.250i·13-s + (0.243 − 0.663i)14-s − 0.757i·15-s + 0.250·16-s + 0.442·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.773 + 0.633i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (643, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.773 + 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.97199 - 1.06103i\)
\(L(\frac12)\) \(\approx\) \(2.97199 - 1.06103i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + iT \)
7 \( 1 + (-0.912 + 2.48i)T \)
23 \( 1 + (4.53 + 1.57i)T \)
good5 \( 1 - 2.93T + 5T^{2} \)
11 \( 1 - 4.68iT - 11T^{2} \)
13 \( 1 + 0.902iT - 13T^{2} \)
17 \( 1 - 1.82T + 17T^{2} \)
19 \( 1 - 5.58T + 19T^{2} \)
29 \( 1 + 9.47T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 1.10iT - 37T^{2} \)
41 \( 1 - 3.09iT - 41T^{2} \)
43 \( 1 + 3.85iT - 43T^{2} \)
47 \( 1 - 0.610iT - 47T^{2} \)
53 \( 1 - 9.23iT - 53T^{2} \)
59 \( 1 + 12.3iT - 59T^{2} \)
61 \( 1 + 7.40T + 61T^{2} \)
67 \( 1 + 4.65iT - 67T^{2} \)
71 \( 1 - 8.37T + 71T^{2} \)
73 \( 1 - 5.51iT - 73T^{2} \)
79 \( 1 + 4.55iT - 79T^{2} \)
83 \( 1 + 1.53T + 83T^{2} \)
89 \( 1 + 3.47T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−9.934899771436775049356435505169, −9.400673243907619213995655281116, −7.82744677578507349572406367473, −7.35808264858749060779834139014, −6.43982563076722773290987416549, −5.57223822158347993668043162427, −4.80527727654261938704576134054, −3.61230670606523580383994197394, −2.23654992034401569302678734988, −1.44191531365675699045935681492, 1.70415801616692373578297891522, 2.81937761087033478582145778095, 3.75718651746706770295007472960, 5.19893834267086013701218943505, 5.67228189905466114787418391518, 6.14557086908977503999558890350, 7.57129545080418752603263770892, 8.622547038992784674132201544806, 9.406682889010764978679149950404, 10.01252629305003009486668059518

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