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Properties

Label 2-2646-63.58-c1-0-5
Degree $2$
Conductor $2646$
Sign $0.933 - 0.359i$
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 + (−1.84 − 3.20i)5-s − 8-s + (1.84 + 3.20i)10-s + (−0.738 + 1.27i)11-s + (1.34 − 2.33i)13-s + 16-s + (3.28 + 5.69i)17-s + (0.444 − 0.769i)19-s + (−1.84 − 3.20i)20-s + (0.738 − 1.27i)22-s + (3.14 + 5.44i)23-s + (−4.34 + 7.52i)25-s + (−1.34 + 2.33i)26-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.827 − 1.43i)5-s − 0.353·8-s + (0.584 + 1.01i)10-s + (−0.222 + 0.385i)11-s + (0.374 − 0.648i)13-s + 0.250·16-s + (0.797 + 1.38i)17-s + (0.101 − 0.176i)19-s + (−0.413 − 0.716i)20-s + (0.157 − 0.272i)22-s + (0.655 + 1.13i)23-s + (−0.868 + 1.50i)25-s + (−0.264 + 0.458i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.933 - 0.359i$
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: \((2,\ 2646,\ (\ :1/2),\ 0.933 - 0.359i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8577779936\)
\(L(\frac12)\) \(\approx\) \(0.8577779936\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.84 + 3.20i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (0.738 - 1.27i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.34 + 2.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.28 - 5.69i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.444 + 0.769i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.14 - 5.44i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.25 + 2.17i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.81T + 31T^{2} \)
37 \( 1 + (1.38 - 2.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.05 - 3.56i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.00618 - 0.0107i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 6.98T + 47T^{2} \)
53 \( 1 + (-1.60 - 2.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 6.90T + 59T^{2} \)
61 \( 1 - 5.73T + 61T^{2} \)
67 \( 1 + 9.46T + 67T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 + (-6.03 - 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + (-2.23 - 3.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.43 - 7.68i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-6.58 - 11.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.830123103264505891954270773962, −8.031579459374549664860884277846, −7.85554617962025310550399889662, −6.82618170946617862777378161625, −5.64158251281370481929085742980, −5.15897811791346525217048507584, −4.01935433216376137538858218889, −3.33063009648671581403321973778, −1.74708074377013536581365554582, −0.900432817667553542524570120713, 0.47222047080707116243635470008, 2.11172579110647583709121723174, 3.12189739730348471685493445591, 3.62850273154343820887494115972, 4.92054037259347881714823580241, 5.98036921839146490589246340919, 6.93030636535928392270096216180, 7.18301329301329664210293235344, 7.990447844054799409697922335064, 8.784521178207250627312644626579

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