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Properties

Label 2-966-161.160-c1-0-18
Degree $2$
Conductor $966$
Sign $0.994 + 0.109i$
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 + 3.74·5-s − i·6-s + (−1.87 − 1.87i)7-s − 8-s − 9-s − 3.74·10-s + i·12-s − 2i·13-s + (1.87 + 1.87i)14-s + 3.74i·15-s + 16-s + 3.74·17-s + 18-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577i·3-s + 0.5·4-s + 1.67·5-s − 0.408i·6-s + (−0.707 − 0.707i)7-s − 0.353·8-s − 0.333·9-s − 1.18·10-s + 0.288i·12-s − 0.554i·13-s + (0.499 + 0.499i)14-s + 0.966i·15-s + 0.250·16-s + 0.907·17-s + 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.994 + 0.109i$
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: \((2,\ 966,\ (\ :1/2),\ 0.994 + 0.109i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46350 - 0.0802585i\)
\(L(\frac12)\) \(\approx\) \(1.46350 - 0.0802585i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.87 + 1.87i)T \)
23 \( 1 + (3 + 3.74i)T \)
good5 \( 1 - 3.74T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 3.74T + 17T^{2} \)
19 \( 1 - 3.74T + 19T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4iT - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 2iT - 41T^{2} \)
43 \( 1 - 11.2iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 3.74iT - 53T^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 3.74iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 - 3.74iT - 79T^{2} \)
83 \( 1 + 7.48T + 83T^{2} \)
89 \( 1 - 3.74T + 89T^{2} \)
97 \( 1 - 7.48T + 97T^{2} \)
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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

−9.970347278011428873344245702998, −9.553783873267372148624898558884, −8.542096522581709937828670534004, −7.55817556886872217289723200015, −6.44753041308674786111570931691, −5.89886067602748568576145236734, −4.90322194971579049509452102493, −3.42439309720576910154652819510, −2.46963275274178490827704973027, −0.990833876426807098094216900052, 1.30756390169699774593462849571, 2.27564002357585539047867005583, 3.21592604369683581621229018348, 5.25872452946625036902448071790, 5.92071392642968751950297531986, 6.57647885599191137458524477858, 7.45720161566668508631493321396, 8.613238624240921713650314802636, 9.275416069701885848509775220201, 9.882370751444589459994497672004

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