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Properties

Label 2-2646-9.4-c1-0-35
Degree $2$
Conductor $2646$
Sign $-0.766 + 0.642i$
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 + (1.5 − 2.59i)5-s + 0.999·8-s − 3·10-s + (−3 − 5.19i)11-s + (1 − 1.73i)13-s + (−0.5 − 0.866i)16-s + 6·17-s + 7·19-s + (1.50 + 2.59i)20-s + (−3 + 5.19i)22-s + (1.5 − 2.59i)23-s + (−2 − 3.46i)25-s − 1.99·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.670 − 1.16i)5-s + 0.353·8-s − 0.948·10-s + (−0.904 − 1.56i)11-s + (0.277 − 0.480i)13-s + (−0.125 − 0.216i)16-s + 1.45·17-s + 1.60·19-s + (0.335 + 0.580i)20-s + (−0.639 + 1.10i)22-s + (0.312 − 0.541i)23-s + (−0.400 − 0.692i)25-s − 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.766 + 0.642i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.702139399\)
\(L(\frac12)\) \(\approx\) \(1.702139399\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1 + 1.73i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4 - 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.581165635121387224328311475920, −8.136548234316799849600785252889, −7.31690035402430237457019156852, −5.85174028708606440916319959394, −5.50568814353730885213975287542, −4.74164071421875689349351870345, −3.35295278102449769477289448994, −2.88597861010257600500156919194, −1.33528395173642461396561539686, −0.72753117013472875710449890119, 1.39210282399091080195609470421, 2.49869977504397099581530802001, 3.36743715795184779857399692561, 4.68678971780584044770460398001, 5.43443644044938719458182217271, 6.17563929073150433115073103314, 7.04032214604342828089044025822, 7.48984138603901845106831580407, 8.123915542011253641777220632514, 9.460277123609880020745082667619

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