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Properties

Label 2-2646-21.20-c1-0-11
Degree $2$
Conductor $2646$
Sign $-0.987 - 0.156i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
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 − 4-s + 1.68·5-s − i·8-s + 1.68i·10-s + 3.32i·11-s + 4.34i·13-s + 16-s − 3.81·17-s − 4.06i·19-s − 1.68·20-s − 3.32·22-s + 4.69i·23-s − 2.16·25-s − 4.34·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + 0.753·5-s − 0.353i·8-s + 0.532i·10-s + 1.00i·11-s + 1.20i·13-s + 0.250·16-s − 0.924·17-s − 0.932i·19-s − 0.376·20-s − 0.709·22-s + 0.979i·23-s − 0.432·25-s − 0.852·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.987 - 0.156i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.987 - 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.195836495\)
\(L(\frac12)\) \(\approx\) \(1.195836495\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 - 1.68T + 5T^{2} \)
11 \( 1 - 3.32iT - 11T^{2} \)
13 \( 1 - 4.34iT - 13T^{2} \)
17 \( 1 + 3.81T + 17T^{2} \)
19 \( 1 + 4.06iT - 19T^{2} \)
23 \( 1 - 4.69iT - 23T^{2} \)
29 \( 1 + 2.50iT - 29T^{2} \)
31 \( 1 - 4.09iT - 31T^{2} \)
37 \( 1 - 6.27T + 37T^{2} \)
41 \( 1 + 5.46T + 41T^{2} \)
43 \( 1 + 10.7T + 43T^{2} \)
47 \( 1 + 0.412T + 47T^{2} \)
53 \( 1 - 9.09iT - 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + 13.9iT - 61T^{2} \)
67 \( 1 + 5.94T + 67T^{2} \)
71 \( 1 - 4.94iT - 71T^{2} \)
73 \( 1 - 5.78iT - 73T^{2} \)
79 \( 1 + 12.9T + 79T^{2} \)
83 \( 1 - 6.04T + 83T^{2} \)
89 \( 1 - 6.31T + 89T^{2} \)
97 \( 1 - 11.6iT - 97T^{2} \)
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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

−9.314372999044402549214278989405, −8.502324157663891511000966723911, −7.54524328761697049251582472586, −6.77465648178328718775815052417, −6.40980120836043768992120594960, −5.30961314212026001919244867451, −4.68754949300026852483034942503, −3.86427039270510793780571712799, −2.44785995005077732786519814193, −1.57975763895580529726796614126, 0.37247281663164837881946481398, 1.66843057924600281786028555903, 2.65387474130560245568844733949, 3.45396759798424035484711397717, 4.43507501987175868030084352186, 5.48578400462126308330028772128, 5.95142406277783802352233307863, 6.87093600327712264450763730226, 8.150600568077795502430574689561, 8.436008033208492609421539776872

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