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Properties

Label 2-966-161.25-c1-0-14
Degree $2$
Conductor $966$
Sign $0.822 - 0.568i$
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  + (0.0475 − 0.998i)2-s + (0.928 + 0.371i)3-s + (−0.995 − 0.0950i)4-s + (0.968 + 3.99i)5-s + (0.415 − 0.909i)6-s + (2.64 − 0.121i)7-s + (−0.142 + 0.989i)8-s + (0.723 + 0.690i)9-s + (4.03 − 0.777i)10-s + (−0.100 − 2.10i)11-s + (−0.888 − 0.458i)12-s + (1.31 + 1.51i)13-s + (0.00406 − 2.64i)14-s + (−0.584 + 4.06i)15-s + (0.981 + 0.189i)16-s + (0.565 + 0.793i)17-s + ⋯
L(s)  = 1  + (0.0336 − 0.706i)2-s + (0.535 + 0.214i)3-s + (−0.497 − 0.0475i)4-s + (0.433 + 1.78i)5-s + (0.169 − 0.371i)6-s + (0.998 − 0.0460i)7-s + (−0.0503 + 0.349i)8-s + (0.241 + 0.230i)9-s + (1.27 − 0.245i)10-s + (−0.0301 − 0.633i)11-s + (−0.256 − 0.132i)12-s + (0.365 + 0.421i)13-s + (0.00108 − 0.707i)14-s + (−0.151 + 1.05i)15-s + (0.245 + 0.0473i)16-s + (0.137 + 0.192i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.822 - 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.05208 + 0.640695i\)
\(L(\frac12)\) \(\approx\) \(2.05208 + 0.640695i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0475 + 0.998i)T \)
3 \( 1 + (-0.928 - 0.371i)T \)
7 \( 1 + (-2.64 + 0.121i)T \)
23 \( 1 + (3.93 + 2.73i)T \)
good5 \( 1 + (-0.968 - 3.99i)T + (-4.44 + 2.29i)T^{2} \)
11 \( 1 + (0.100 + 2.10i)T + (-10.9 + 1.04i)T^{2} \)
13 \( 1 + (-1.31 - 1.51i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-0.565 - 0.793i)T + (-5.56 + 16.0i)T^{2} \)
19 \( 1 + (3.47 - 4.88i)T + (-6.21 - 17.9i)T^{2} \)
29 \( 1 + (-4.16 + 9.12i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-5.57 - 4.38i)T + (7.30 + 30.1i)T^{2} \)
37 \( 1 + (0.324 + 0.309i)T + (1.76 + 36.9i)T^{2} \)
41 \( 1 + (8.81 + 2.58i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (-0.808 - 5.62i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (-6.58 - 11.4i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.89 + 5.46i)T + (-41.6 + 32.7i)T^{2} \)
59 \( 1 + (9.57 - 1.84i)T + (54.7 - 21.9i)T^{2} \)
61 \( 1 + (-6.00 + 2.40i)T + (44.1 - 42.0i)T^{2} \)
67 \( 1 + (-8.09 + 4.17i)T + (38.8 - 54.5i)T^{2} \)
71 \( 1 + (-1.09 - 0.705i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-1.20 - 0.115i)T + (71.6 + 13.8i)T^{2} \)
79 \( 1 + (-2.64 + 7.65i)T + (-62.0 - 48.8i)T^{2} \)
83 \( 1 + (-10.6 + 3.13i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-1.88 + 1.48i)T + (20.9 - 86.4i)T^{2} \)
97 \( 1 + (-8.90 - 2.61i)T + (81.6 + 52.4i)T^{2} \)
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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

−10.39417756999409417082672212962, −9.542147594066727062047614084169, −8.292921416826856164172354446170, −7.931116068980979041792746625542, −6.56339185099324322060162139153, −5.93293666278761634246903389442, −4.46120038058142100444110604438, −3.59859841611368221305116010351, −2.62396796089727732679106543735, −1.79101471928237183756246740740, 1.01219491857585580045218002228, 2.13787090867848978616162592447, 4.00243707848564250279226819358, 4.89457699826594173015833323295, 5.34127598894972168722584701949, 6.57076437881988408541445480975, 7.64610376650376587057143035787, 8.426226071078713868148573189103, 8.756875268818466099706907014899, 9.590085844997258226596240932626

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