Properties

Label 2-2646-21.20-c1-0-35
Degree $2$
Conductor $2646$
Sign $-0.654 + 0.755i$
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 + i·8-s + 3i·11-s − 3.46i·13-s + 16-s + 3·22-s − 5·25-s − 3.46·26-s − 9i·29-s + 1.73i·31-s i·32-s − 8·37-s + 10.3·41-s + 4·43-s − 3i·44-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.353i·8-s + 0.904i·11-s − 0.960i·13-s + 0.250·16-s + 0.639·22-s − 25-s − 0.679·26-s − 1.67i·29-s + 0.311i·31-s − 0.176i·32-s − 1.31·37-s + 1.62·41-s + 0.609·43-s − 0.452i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.654 + 0.755i$
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: \((2,\ 2646,\ (\ :1/2),\ -0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.223273165\)
\(L(\frac12)\) \(\approx\) \(1.223273165\)
\(L(\frac{3}{2})\) 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 \( 1 + 5T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 - 1.73iT - 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + 6iT - 53T^{2} \)
59 \( 1 - 5.19T + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 12iT - 71T^{2} \)
73 \( 1 + 5.19iT - 73T^{2} \)
79 \( 1 + 13T + 79T^{2} \)
83 \( 1 + 5.19T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 8.66iT - 97T^{2} \)
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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

−8.653419652234770887780091947343, −7.86006074265710689335683392557, −7.26420727806180690533881199717, −6.11027004683910583175952793596, −5.40148200973004817018802523135, −4.45494770468951284887059641161, −3.73739690963575306371655323228, −2.66734810818300502285849989552, −1.84395497427287638255664399873, −0.43902315514239629342500150671, 1.17415559689160526998408073769, 2.56096117314947496579523076061, 3.73850560760247235730777413811, 4.38317418358235772484216556305, 5.54243244906231032844768619512, 5.93547577587174152902369081047, 7.00883344249636638823367877524, 7.40446607665820511407530553811, 8.533299262418267129049752013481, 8.871580380558174908287251089072

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