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Properties

Label 2-2646-63.25-c1-0-27
Degree $2$
Conductor $2646$
Sign $0.374 + 0.927i$
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  − 2-s + 4-s + (1.72 − 2.98i)5-s − 8-s + (−1.72 + 2.98i)10-s + (1 + 1.73i)11-s + (2.44 + 4.24i)13-s + 16-s + (1 − 1.73i)17-s + (−3.72 − 6.45i)19-s + (1.72 − 2.98i)20-s + (−1 − 1.73i)22-s + (−0.5 + 0.866i)23-s + (−3.44 − 5.97i)25-s + (−2.44 − 4.24i)26-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (0.771 − 1.33i)5-s − 0.353·8-s + (−0.545 + 0.944i)10-s + (0.301 + 0.522i)11-s + (0.679 + 1.17i)13-s + 0.250·16-s + (0.242 − 0.420i)17-s + (−0.854 − 1.48i)19-s + (0.385 − 0.667i)20-s + (−0.213 − 0.369i)22-s + (−0.104 + 0.180i)23-s + (−0.689 − 1.19i)25-s + (−0.480 − 0.832i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.550537252\)
\(L(\frac12)\) \(\approx\) \(1.550537252\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 2.98i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.44 - 4.24i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.72 + 6.45i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.44 + 2.51i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + (-3.89 - 6.75i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.89 + 8.48i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.44 + 2.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + (0.550 - 0.953i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 3.10T + 67T^{2} \)
71 \( 1 + 9.89T + 71T^{2} \)
73 \( 1 + (1.44 - 2.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 7.89T + 79T^{2} \)
83 \( 1 + (-1 + 1.73i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.55 + 6.14i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.44 + 5.97i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.804001421743130110348537491958, −8.329417260418570192286462602666, −7.13699663591839246767926655833, −6.54136463651771662881226170689, −5.69329758837126645876684151782, −4.74414521228521980884500179202, −4.14325057561978835544812227956, −2.55523631298075331914249302750, −1.70171082188549137045509033311, −0.73853774495165052939215764754, 1.14144767430640714524353491832, 2.30447416181606719408997646369, 3.12423846895018352391392032411, 3.92337405648942411085544281506, 5.53497068446004951374084479233, 6.17882107204149486263170419547, 6.53182730875724569718458218700, 7.65229147837121448146437712437, 8.202827016571675421910852986473, 8.963693927397224669235413263719

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