Properties

Label 2-966-21.5-c1-0-55
Degree $2$
Conductor $966$
Sign $0.553 + 0.832i$
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  + (0.866 + 0.5i)2-s + (1.5 − 0.866i)3-s + (0.499 + 0.866i)4-s + (1.73 − 3i)5-s + 1.73·6-s + (−2.5 − 0.866i)7-s + 0.999i·8-s + (1.5 − 2.59i)9-s + (3 − 1.73i)10-s + (−2.59 + 1.5i)11-s + (1.49 + 0.866i)12-s − 3.46i·13-s + (−1.73 − 2i)14-s − 6i·15-s + (−0.5 + 0.866i)16-s + (0.866 + 1.5i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.866 − 0.499i)3-s + (0.249 + 0.433i)4-s + (0.774 − 1.34i)5-s + 0.707·6-s + (−0.944 − 0.327i)7-s + 0.353i·8-s + (0.5 − 0.866i)9-s + (0.948 − 0.547i)10-s + (−0.783 + 0.452i)11-s + (0.433 + 0.250i)12-s − 0.960i·13-s + (−0.462 − 0.534i)14-s − 1.54i·15-s + (−0.125 + 0.216i)16-s + (0.210 + 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.68108 - 1.43664i\)
\(L(\frac12)\) \(\approx\) \(2.68108 - 1.43664i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
good5 \( 1 + (-1.73 + 3i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + (-0.866 - 1.5i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-6 - 3.46i)T + (9.5 + 16.4i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (-3 + 1.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1 + 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (4.33 - 7.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.3 - 6i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.92 - 12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12 + 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9iT - 71T^{2} \)
73 \( 1 + (-7.5 + 4.33i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 + (3.46 - 6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.46iT - 97T^{2} \)
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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

−9.688283003029621609887291664318, −9.089661143606812505992883229061, −7.88138994095482360096174369650, −7.67584171890563718906824033329, −6.27069278489263540851888730706, −5.66176365005791203957000786204, −4.62873011110366232595882637017, −3.49900781072499852421896034421, −2.51616565001459956290848056573, −1.10342726057049501788041806710, 2.10168030921113068437880000780, 3.01422550492283049541736641957, 3.34615779536844302767168017299, 4.82840659399795029565920689336, 5.77893279322580307327085732682, 6.74322152754312084948694677583, 7.38024207767404975473705119753, 8.743576172040483476963821310737, 9.747491958664350789777048145926, 9.907280419103882669862681700751

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