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Properties

Label 2-3264-17.16-c1-0-14
Degree $2$
Conductor $3264$
Sign $-0.874 + 0.485i$
Analytic cond. $26.0631$
Root an. cond. $5.10521$
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·3-s + 3.60i·5-s + 4i·7-s − 9-s − 3i·11-s + 13-s − 3.60·15-s + (2 + 3.60i)17-s − 3.60·19-s − 4·21-s + 7i·23-s − 7.99·25-s − i·27-s − 7.21i·29-s − 2i·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.61i·5-s + 1.51i·7-s − 0.333·9-s − 0.904i·11-s + 0.277·13-s − 0.930·15-s + (0.485 + 0.874i)17-s − 0.827·19-s − 0.872·21-s + 1.45i·23-s − 1.59·25-s − 0.192i·27-s − 1.33i·29-s − 0.359i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3264\)    =    \(2^{6} \cdot 3 \cdot 17\)
Sign: $-0.874 + 0.485i$
Analytic conductor: \(26.0631\)
Root analytic conductor: \(5.10521\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3264} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3264,\ (\ :1/2),\ -0.874 + 0.485i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.330071879\)
\(L(\frac12)\) \(\approx\) \(1.330071879\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - iT \)
17 \( 1 + (-2 - 3.60i)T \)
good5 \( 1 - 3.60iT - 5T^{2} \)
7 \( 1 - 4iT - 7T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 - T + 13T^{2} \)
19 \( 1 + 3.60T + 19T^{2} \)
23 \( 1 - 7iT - 23T^{2} \)
29 \( 1 + 7.21iT - 29T^{2} \)
31 \( 1 + 2iT - 31T^{2} \)
37 \( 1 - 7.21iT - 37T^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 - 3.60T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 4T + 53T^{2} \)
59 \( 1 + 14.4T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 8iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 4iT - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 7.21iT - 97T^{2} \)
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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

−9.177282304622525289143865251569, −8.241388044378810139239803117758, −7.82206068201367802050228605867, −6.52969013575674232711405333101, −6.03409336924196122506041196585, −5.60916359743494299583325820428, −4.32032739669995161574643406801, −3.30205654315505312699761837252, −2.90135181571029461935264849556, −1.88357692654228484280871637079, 0.43150517976570745047377109465, 1.17515422458467045344967637243, 2.18417488120713193625560277486, 3.65516187286995926704062629032, 4.47591458901344526899460604475, 4.93845446055131428952064951371, 5.92851573462344790543508802686, 7.02975067110718443500894949193, 7.33017957664001774466043605277, 8.241668730599098488442726031028

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