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Properties

Label 2-966-21.20-c1-0-2
Degree $2$
Conductor $966$
Sign $0.365 - 0.930i$
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  − i·2-s + (0.375 − 1.69i)3-s − 4-s + 0.513·5-s + (−1.69 − 0.375i)6-s + (−2.61 − 0.410i)7-s + i·8-s + (−2.71 − 1.26i)9-s − 0.513i·10-s + 3.12i·11-s + (−0.375 + 1.69i)12-s + 5.94i·13-s + (−0.410 + 2.61i)14-s + (0.192 − 0.868i)15-s + 16-s − 5.76·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.216 − 0.976i)3-s − 0.5·4-s + 0.229·5-s + (−0.690 − 0.153i)6-s + (−0.987 − 0.155i)7-s + 0.353i·8-s + (−0.906 − 0.423i)9-s − 0.162i·10-s + 0.942i·11-s + (−0.108 + 0.488i)12-s + 1.64i·13-s + (−0.109 + 0.698i)14-s + (0.0497 − 0.224i)15-s + 0.250·16-s − 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.365 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.365 - 0.930i$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{966} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 0.365 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.237832 + 0.162117i\)
\(L(\frac12)\) \(\approx\) \(0.237832 + 0.162117i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.375 + 1.69i)T \)
7 \( 1 + (2.61 + 0.410i)T \)
23 \( 1 - iT \)
good5 \( 1 - 0.513T + 5T^{2} \)
11 \( 1 - 3.12iT - 11T^{2} \)
13 \( 1 - 5.94iT - 13T^{2} \)
17 \( 1 + 5.76T + 17T^{2} \)
19 \( 1 + 0.598iT - 19T^{2} \)
29 \( 1 + 7.46iT - 29T^{2} \)
31 \( 1 - 7.72iT - 31T^{2} \)
37 \( 1 + 1.50T + 37T^{2} \)
41 \( 1 - 1.20T + 41T^{2} \)
43 \( 1 - 5.20T + 43T^{2} \)
47 \( 1 + 6.27T + 47T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 0.414iT - 61T^{2} \)
67 \( 1 + 6.23T + 67T^{2} \)
71 \( 1 + 4.92iT - 71T^{2} \)
73 \( 1 - 3.88iT - 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + 3.08T + 89T^{2} \)
97 \( 1 - 4.96iT - 97T^{2} \)
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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

−10.04218229274810281040573494663, −9.328071251799172342075030408013, −8.801412238533210495502212149803, −7.56545926911901461079542806855, −6.73086155226157942912991737948, −6.19494961678298398507659364275, −4.70311174795749499596269725487, −3.74147774688116252103073907113, −2.44691762884650672136669093662, −1.73908377729041418303650884699, 0.12134773257277399074638798049, 2.75254483972403503043014664747, 3.55130588785962636132050227433, 4.62171919730931241734415266822, 5.81900144799570032388685630904, 6.02606306914314943068523542625, 7.40427729487182563763830821704, 8.359745700355246639167859507485, 9.005450547230505158370709394317, 9.715075554670141747001426290143

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