Properties

Label 2-2646-9.7-c1-0-30
Degree $2$
Conductor $2646$
Sign $-0.755 + 0.655i$
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  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.880 − 1.52i)5-s − 0.999·8-s − 1.76·10-s + (3.06 − 5.30i)11-s + (0.380 + 0.658i)13-s + (−0.5 + 0.866i)16-s + 6.84·17-s + 1.94·19-s + (−0.880 + 1.52i)20-s + (−3.06 − 5.30i)22-s + (−0.210 − 0.364i)23-s + (0.949 − 1.64i)25-s + 0.760·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.393 − 0.681i)5-s − 0.353·8-s − 0.556·10-s + (0.923 − 1.59i)11-s + (0.105 + 0.182i)13-s + (−0.125 + 0.216i)16-s + 1.65·17-s + 0.445·19-s + (−0.196 + 0.340i)20-s + (−0.652 − 1.13i)22-s + (−0.0438 − 0.0760i)23-s + (0.189 − 0.328i)25-s + 0.149·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.755 + 0.655i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (883, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.755 + 0.655i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.073699851\)
\(L(\frac12)\) \(\approx\) \(2.073699851\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.880 + 1.52i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.06 + 5.30i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.380 - 0.658i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6.84T + 17T^{2} \)
19 \( 1 - 1.94T + 19T^{2} \)
23 \( 1 + (0.210 + 0.364i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.732 - 1.26i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.85 - 6.67i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.88T + 37T^{2} \)
41 \( 1 + (3.47 + 6.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.33 + 7.49i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.830 - 1.43i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.225T + 53T^{2} \)
59 \( 1 + (0.993 + 1.72i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.17 - 8.96i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.39 + 5.87i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 0.306T + 73T^{2} \)
79 \( 1 + (-6.72 + 11.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.56 - 2.70i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.60T + 89T^{2} \)
97 \( 1 + (-1.81 + 3.14i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.744958692740211742332238320213, −8.018369382882049737264463017344, −6.99834628928488417106165383768, −5.98757561741313068117319368217, −5.42810906699085949056624312318, −4.49742349757444043070551312955, −3.56490555905005668295026486914, −3.08918064842860420899952163914, −1.47249698233592522459440435490, −0.69207241045606448748774346327, 1.38293477073359602657768335207, 2.79366350042823939024729892943, 3.64716044219094534818172631278, 4.41748851118740803216726022807, 5.28397149684227859553418196282, 6.21101163400009923235343344151, 6.88672036168087404712267337381, 7.62205681670613326806296059426, 7.973286000240538383400219274213, 9.262416086435990741992060410615

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