Properties

Label 2-2646-1.1-c1-0-33
Degree $2$
Conductor $2646$
Sign $-1$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 1.58·5-s − 8-s + 1.58·10-s − 11-s + 4.24·13-s + 16-s − 2.82·17-s + 0.171·19-s − 1.58·20-s + 22-s − 3.24·23-s − 2.48·25-s − 4.24·26-s + 8.24·29-s − 1.24·31-s − 32-s + 2.82·34-s − 3.24·37-s − 0.171·38-s + 1.58·40-s − 11.8·41-s + 10.4·43-s − 44-s + 3.24·46-s − 0.343·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.709·5-s − 0.353·8-s + 0.501·10-s − 0.301·11-s + 1.17·13-s + 0.250·16-s − 0.685·17-s + 0.0393·19-s − 0.354·20-s + 0.213·22-s − 0.676·23-s − 0.497·25-s − 0.832·26-s + 1.53·29-s − 0.223·31-s − 0.176·32-s + 0.485·34-s − 0.533·37-s − 0.0278·38-s + 0.250·40-s − 1.84·41-s + 1.59·43-s − 0.150·44-s + 0.478·46-s − 0.0500·47-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 - 4.24T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 0.171T + 19T^{2} \)
23 \( 1 + 3.24T + 23T^{2} \)
29 \( 1 - 8.24T + 29T^{2} \)
31 \( 1 + 1.24T + 31T^{2} \)
37 \( 1 + 3.24T + 37T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 + 0.343T + 47T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + 0.343T + 59T^{2} \)
61 \( 1 + 9.17T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 7.24T + 71T^{2} \)
73 \( 1 - 8.82T + 73T^{2} \)
79 \( 1 - 10.4T + 79T^{2} \)
83 \( 1 + 15.8T + 83T^{2} \)
89 \( 1 + 14.6T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−8.407402061190923619451755323100, −7.962035694241662108664010302925, −7.03639357661927983510146439714, −6.36881981196400468140037820839, −5.50106927532037167917415566171, −4.35365403185281411574788853838, −3.60216949801363332690105970281, −2.54995284564172312716117646968, −1.35050129198020884482127952700, 0, 1.35050129198020884482127952700, 2.54995284564172312716117646968, 3.60216949801363332690105970281, 4.35365403185281411574788853838, 5.50106927532037167917415566171, 6.36881981196400468140037820839, 7.03639357661927983510146439714, 7.962035694241662108664010302925, 8.407402061190923619451755323100

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