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Properties

Label 2-2646-63.58-c1-0-6
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 − 8-s + (−1.5 + 2.59i)11-s + (1 − 1.73i)13-s + 16-s + (1.5 + 2.59i)17-s + (−0.5 + 0.866i)19-s + (1.5 − 2.59i)22-s + (−3 − 5.19i)23-s + (2.5 − 4.33i)25-s + (−1 + 1.73i)26-s + (3 + 5.19i)29-s + 4·31-s − 32-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.353·8-s + (−0.452 + 0.783i)11-s + (0.277 − 0.480i)13-s + 0.250·16-s + (0.363 + 0.630i)17-s + (−0.114 + 0.198i)19-s + (0.319 − 0.553i)22-s + (−0.625 − 1.08i)23-s + (0.5 − 0.866i)25-s + (−0.196 + 0.339i)26-s + (0.557 + 0.964i)29-s + 0.718·31-s − 0.176·32-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: $\chi_{2646} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.580 - 0.814i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.102416577\)
\(L(\frac12)\) \(\approx\) \(1.102416577\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.5 - 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + (-6 - 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 5T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
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

−8.819306042559149366712982146397, −8.244534904495556194671033432831, −7.67802713463572480188146163166, −6.68750520259703005942461616312, −6.14458805906463879378489249533, −5.08160046829492507545557193059, −4.23683440606593792972655425284, −3.06181085702661795380378601278, −2.19041502823851050101064873308, −0.985264633737649693461087616249, 0.55812430525701541304521890626, 1.80330345108160920884028359169, 2.91505730668241754735582627927, 3.73511767660130153173940175456, 4.96318117081020586353393678419, 5.74136872007093498487334764758, 6.58086326082124629834983295199, 7.31805930353675071367748332626, 8.149078322124161705355530980907, 8.635057422957178197069485948737

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