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Properties

Label 2-2646-63.16-c1-0-5
Degree $2$
Conductor $2646$
Sign $-0.175 - 0.984i$
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 + 0.593·5-s + 0.999·8-s + (−0.296 − 0.514i)10-s + 0.593·11-s + (1.25 + 2.17i)13-s + (−0.5 − 0.866i)16-s + (1.46 + 2.52i)17-s + (−2.69 + 4.66i)19-s + (−0.296 + 0.514i)20-s + (−0.296 − 0.514i)22-s − 4.46·23-s − 4.64·25-s + (1.25 − 2.17i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 0.265·5-s + 0.353·8-s + (−0.0938 − 0.162i)10-s + 0.178·11-s + (0.348 + 0.603i)13-s + (−0.125 − 0.216i)16-s + (0.354 + 0.613i)17-s + (−0.617 + 1.06i)19-s + (−0.0663 + 0.114i)20-s + (−0.0632 − 0.109i)22-s − 0.930·23-s − 0.929·25-s + (0.246 − 0.427i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.175 - 0.984i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.175 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6884494532\)
\(L(\frac12)\) \(\approx\) \(0.6884494532\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.593T + 5T^{2} \)
11 \( 1 - 0.593T + 11T^{2} \)
13 \( 1 + (-1.25 - 2.17i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.46 - 2.52i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.69 - 4.66i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 4.46T + 23T^{2} \)
29 \( 1 + (-3.09 + 5.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.93 - 6.81i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.136 + 0.236i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.58 - 9.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.08 + 10.5i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.02 + 6.97i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.32 - 7.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.32 + 5.75i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.956 + 1.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 + (3.95 + 6.85i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.62 - 8.00i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.85 + 6.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.21 - 10.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.86 - 10.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.211343969721315373769634300418, −8.177368110730402319771363267458, −8.004723755064993841700854457100, −6.65704188780079613945160641449, −6.13387846949106982810454875505, −5.09896050266817734104276647099, −4.02935023614271208106957541005, −3.49369834366745149090777647308, −2.14046664033809257741214564003, −1.48248403363810641020347091196, 0.24790360963850521169569613457, 1.63946970784245799611940640114, 2.81836439384658533318556635292, 3.94711289109007552228311363075, 4.88553887340049754263921461611, 5.69573612586818489075352842846, 6.35572699595174127366338540263, 7.16095988197719403478849570789, 7.911577450235593896980349794058, 8.563240843114823842717470654543

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