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Properties

Label 2-2646-63.20-c1-0-11
Degree $2$
Conductor $2646$
Sign $0.435 - 0.900i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−0.483 − 0.837i)5-s − 0.999i·8-s + 0.967i·10-s + (4.82 + 2.78i)11-s + (−3.76 + 2.17i)13-s + (−0.5 + 0.866i)16-s + 3.94·17-s + 4.46i·19-s + (0.483 − 0.837i)20-s + (−2.78 − 4.82i)22-s + (2.29 − 1.32i)23-s + (2.03 − 3.51i)25-s + 4.35·26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.216 − 0.374i)5-s − 0.353i·8-s + 0.305i·10-s + (1.45 + 0.840i)11-s + (−1.04 + 0.603i)13-s + (−0.125 + 0.216i)16-s + 0.956·17-s + 1.02i·19-s + (0.108 − 0.187i)20-s + (−0.594 − 1.02i)22-s + (0.479 − 0.276i)23-s + (0.406 − 0.703i)25-s + 0.853·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.435 - 0.900i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.435 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.025667385\)
\(L(\frac12)\) \(\approx\) \(1.025667385\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.483 + 0.837i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-4.82 - 2.78i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.76 - 2.17i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.94T + 17T^{2} \)
19 \( 1 - 4.46iT - 19T^{2} \)
23 \( 1 + (-2.29 + 1.32i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.61 + 2.66i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.34 - 3.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.487T + 37T^{2} \)
41 \( 1 + (0.0818 + 0.141i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.35 - 7.53i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.74 - 8.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 2.01iT - 53T^{2} \)
59 \( 1 + (-0.836 - 1.44i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.47 + 2.58i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.72 - 4.71i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.64iT - 71T^{2} \)
73 \( 1 - 2.48iT - 73T^{2} \)
79 \( 1 + (2.30 - 3.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.20 + 7.29i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 4.11T + 89T^{2} \)
97 \( 1 + (-10.2 - 5.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.159451559691203053496452854792, −8.314451554830895380709671348249, −7.53084755668234003430526660288, −6.91719888198213652869303944651, −6.07519657627963055648240287333, −4.89197872124756533405462183863, −4.17672205267840222313816491030, −3.29757186660729849687713409589, −2.02502353949336219907902175738, −1.20869949135131539920316883346, 0.45357552346232670139221274578, 1.68897135297855767129434182352, 3.04719476896605004050030277904, 3.71449491057450864024735394745, 5.06176046545525268251549240413, 5.66008771940237626032409277220, 6.70365255137706407218312075501, 7.19643361911762601074611649084, 7.87363595194528475020720077349, 8.902185027721625540567747731452

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