Properties

Label 2-2646-21.20-c1-0-44
Degree $2$
Conductor $2646$
Sign $-0.409 + 0.912i$
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  i·2-s − 4-s + 4.29·5-s + i·8-s − 4.29i·10-s − 5.20i·11-s + 2.81i·13-s + 16-s − 4.02·17-s − 1.77i·19-s − 4.29·20-s − 5.20·22-s − 7.70i·23-s + 13.4·25-s + 2.81·26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 1.92·5-s + 0.353i·8-s − 1.35i·10-s − 1.56i·11-s + 0.780i·13-s + 0.250·16-s − 0.976·17-s − 0.408i·19-s − 0.960·20-s − 1.10·22-s − 1.60i·23-s + 2.69·25-s + 0.551·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.409 + 0.912i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.409 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.250552624\)
\(L(\frac12)\) \(\approx\) \(2.250552624\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - 4.29T + 5T^{2} \)
11 \( 1 + 5.20iT - 11T^{2} \)
13 \( 1 - 2.81iT - 13T^{2} \)
17 \( 1 + 4.02T + 17T^{2} \)
19 \( 1 + 1.77iT - 19T^{2} \)
23 \( 1 + 7.70iT - 23T^{2} \)
29 \( 1 + 5.02iT - 29T^{2} \)
31 \( 1 + 1.25iT - 31T^{2} \)
37 \( 1 + 9.17T + 37T^{2} \)
41 \( 1 + 5.29T + 41T^{2} \)
43 \( 1 + 4.53T + 43T^{2} \)
47 \( 1 - 7.59T + 47T^{2} \)
53 \( 1 - 0.167iT - 53T^{2} \)
59 \( 1 - 5.99T + 59T^{2} \)
61 \( 1 + 7.79iT - 61T^{2} \)
67 \( 1 - 11.3T + 67T^{2} \)
71 \( 1 - 0.539iT - 71T^{2} \)
73 \( 1 + 3.80iT - 73T^{2} \)
79 \( 1 - 1.99T + 79T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 + 1.29T + 89T^{2} \)
97 \( 1 - 17.2iT - 97T^{2} \)
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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

−8.928736557507795228409104613932, −8.268465212881685289852722081052, −6.61498821844749474753384913282, −6.42587126589281191485748286441, −5.43809478442450203027924724832, −4.77310007069655323816960309331, −3.60875110622310436450360759061, −2.49906279194693927604297665411, −2.01220939405914222749702859628, −0.71373519968225971097398030494, 1.50385856239971373148209771966, 2.21159696666103307182493689264, 3.46880933102432653914702727018, 4.81011790684432774527817334742, 5.30514431903632185404175240185, 5.96076672434562165372713088041, 6.92475025679073581234335122240, 7.21499043621272590174782651890, 8.490916338928333912070229827380, 9.085892020123595377085958864535

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