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Properties

Label 2-966-7.4-c1-0-15
Degree $2$
Conductor $966$
Sign $-0.198 + 0.980i$
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  + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·6-s + (−2.62 − 0.358i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (1.5 − 2.59i)11-s + (−0.499 − 0.866i)12-s − 1.41·13-s + (1 + 2.44i)14-s − 0.999·15-s + (−0.5 − 0.866i)16-s + (−0.707 + 1.22i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.223 + 0.387i)5-s + 0.408·6-s + (−0.990 − 0.135i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.452 − 0.783i)11-s + (−0.144 − 0.249i)12-s − 0.392·13-s + (0.267 + 0.654i)14-s − 0.258·15-s + (−0.125 − 0.216i)16-s + (−0.171 + 0.297i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.450638 - 0.550802i\)
\(L(\frac12)\) \(\approx\) \(0.450638 - 0.550802i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.62 + 0.358i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (-0.5 - 0.866i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.292 - 0.507i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 + 9.24T + 29T^{2} \)
31 \( 1 + (-3.32 + 5.76i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.53 + 4.39i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.2T + 41T^{2} \)
43 \( 1 - 3.65T + 43T^{2} \)
47 \( 1 + (6.65 + 11.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.91 + 6.77i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.62 + 8.00i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.82 + 13.5i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + (-2.82 + 4.89i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.20 + 2.09i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + (-5.36 - 9.29i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.41T + 97T^{2} \)
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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

−9.754498837625337846517425415696, −9.313487723417173909486896397162, −8.357624177720393559858776672804, −7.21884720926094081766596283022, −6.32120569169977036863396451410, −5.50581256005384880812755183741, −4.09544115576368824465427532432, −3.42107707822748857855838296284, −2.29490781854906930480249292512, −0.41796732134421822642017607744, 1.26940145688420037913210240131, 2.71901848258866962937637022354, 4.24118741443182359269470920858, 5.29317422911654618217618339367, 6.09097071538205884548705388954, 6.99734403473974909215378474809, 7.43788129136831696033791254681, 8.696618274312697202181003820211, 9.359164813726855587170163971127, 9.948939470589015210106956596588

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