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Properties

Label 2-966-161.145-c1-0-28
Degree $2$
Conductor $966$
Sign $0.353 + 0.935i$
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  + (0.327 + 0.945i)2-s + (0.458 − 0.888i)3-s + (−0.786 + 0.618i)4-s + (−0.000133 − 1.27e−5i)5-s + (0.989 + 0.142i)6-s + (1.26 − 2.32i)7-s + (−0.841 − 0.540i)8-s + (−0.580 − 0.814i)9-s + (−3.16e−5 − 0.000130i)10-s + (−2.18 − 0.757i)11-s + (0.189 + 0.981i)12-s + (1.48 − 5.06i)13-s + (2.61 + 0.430i)14-s + (−7.26e−5 + 0.000113i)15-s + (0.235 − 0.971i)16-s + (−4.72 + 1.89i)17-s + ⋯
L(s)  = 1  + (0.231 + 0.668i)2-s + (0.264 − 0.513i)3-s + (−0.393 + 0.309i)4-s + (−5.98e−5 − 5.71e−6i)5-s + (0.404 + 0.0580i)6-s + (0.476 − 0.879i)7-s + (−0.297 − 0.191i)8-s + (−0.193 − 0.271i)9-s + (−1.00e−5 − 4.12e−5i)10-s + (−0.659 − 0.228i)11-s + (0.0546 + 0.283i)12-s + (0.412 − 1.40i)13-s + (0.697 + 0.114i)14-s + (−1.87e−5 + 2.91e−5i)15-s + (0.0589 − 0.242i)16-s + (−1.14 + 0.458i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.353 + 0.935i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.353 + 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23228 - 0.851538i\)
\(L(\frac12)\) \(\approx\) \(1.23228 - 0.851538i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.327 - 0.945i)T \)
3 \( 1 + (-0.458 + 0.888i)T \)
7 \( 1 + (-1.26 + 2.32i)T \)
23 \( 1 + (-3.59 + 3.17i)T \)
good5 \( 1 + (0.000133 + 1.27e-5i)T + (4.90 + 0.946i)T^{2} \)
11 \( 1 + (2.18 + 0.757i)T + (8.64 + 6.79i)T^{2} \)
13 \( 1 + (-1.48 + 5.06i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (4.72 - 1.89i)T + (12.3 - 11.7i)T^{2} \)
19 \( 1 + (2.45 + 0.981i)T + (13.7 + 13.1i)T^{2} \)
29 \( 1 + (0.0478 - 0.332i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-0.479 - 0.0228i)T + (30.8 + 2.94i)T^{2} \)
37 \( 1 + (-7.64 + 5.44i)T + (12.1 - 34.9i)T^{2} \)
41 \( 1 + (5.37 - 2.45i)T + (26.8 - 30.9i)T^{2} \)
43 \( 1 + (-1.37 - 2.14i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + (-1.41 - 0.817i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-7.60 + 7.97i)T + (-2.52 - 52.9i)T^{2} \)
59 \( 1 + (0.946 - 0.229i)T + (52.4 - 27.0i)T^{2} \)
61 \( 1 + (-5.85 + 3.01i)T + (35.3 - 49.6i)T^{2} \)
67 \( 1 + (-0.312 + 1.62i)T + (-62.2 - 24.9i)T^{2} \)
71 \( 1 + (-2.20 + 2.54i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-8.53 - 10.8i)T + (-17.2 + 70.9i)T^{2} \)
79 \( 1 + (-1.82 - 1.90i)T + (-3.75 + 78.9i)T^{2} \)
83 \( 1 + (2.17 - 4.75i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (0.354 + 7.43i)T + (-88.5 + 8.45i)T^{2} \)
97 \( 1 + (7.72 + 16.9i)T + (-63.5 + 73.3i)T^{2} \)
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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

−9.844826822037738501795296542370, −8.527858897796249121780938411663, −8.189904031409061150353001319502, −7.35697517102502937576929113855, −6.53339332081747900768592466921, −5.63949366434759152310780688645, −4.61584908967294455396204888985, −3.63626353582881997852220951337, −2.37563692138627286770660352262, −0.60960658181908175762143311501, 1.86125002293713048936866202005, 2.65248536768127760166462667133, 3.98215523124711468455405989963, 4.72121072130280153922164416357, 5.59987376765114131060944016501, 6.67163144368746580113856188491, 7.909834612471419749976480092387, 8.871795200637956005481096747344, 9.262701361255510218846968868480, 10.20005614045243921523853866821

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