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Properties

Label 2-2646-63.25-c1-0-31
Degree $2$
Conductor $2646$
Sign $0.580 + 0.814i$
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 + 1.73i)5-s + 8-s + (−1 + 1.73i)10-s + (0.5 + 0.866i)11-s + (−3 − 5.19i)13-s + 16-s + (2.5 − 4.33i)17-s + (−3.5 − 6.06i)19-s + (−1 + 1.73i)20-s + (0.5 + 0.866i)22-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + (−3 − 5.19i)26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.447 + 0.774i)5-s + 0.353·8-s + (−0.316 + 0.547i)10-s + (0.150 + 0.261i)11-s + (−0.832 − 1.44i)13-s + 0.250·16-s + (0.606 − 1.05i)17-s + (−0.802 − 1.39i)19-s + (−0.223 + 0.387i)20-s + (0.106 + 0.184i)22-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + (−0.588 − 1.01i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.580 + 0.814i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.580 + 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.238293608\)
\(L(\frac12)\) \(\approx\) \(2.238293608\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3 + 5.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.5 + 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.5 + 6.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2 - 3.46i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 7T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + 6T + 79T^{2} \)
83 \( 1 + (8 - 13.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.5 - 4.33i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.605883999802532178731957389133, −7.77931901657701262305213377583, −6.98116186406118996901187215024, −6.71160211739112702738620937727, −5.34229888889911994662927712937, −4.98605236920549854540505505085, −3.87465620460471598696718811595, −2.93773597615254736522531702806, −2.46873780427177624427120869078, −0.59523808368005027353619726588, 1.29618045355674907519473690409, 2.28054833530608926522103083145, 3.62518222537799671455367773040, 4.21424714782467516569326268032, 4.89783466270173034517463189919, 5.88423158706739244196362519730, 6.49450195190736478250962917695, 7.46953214515554653338834033136, 8.184883916916409380258534138285, 8.824537020284113034576864295785

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