Properties

Label 2-28e2-7.6-c2-0-16
Degree $2$
Conductor $784$
Sign $0.755 - 0.654i$
Analytic cond. $21.3624$
Root an. cond. $4.62195$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.717i·3-s + 6.63i·5-s + 8.48·9-s + 4.75·11-s − 15.2i·13-s + 4.75·15-s − 3.76i·17-s + 4.18i·19-s + 27.7·23-s − 18.9·25-s − 12.5i·27-s + 3.51·29-s + 48.8i·31-s − 3.41i·33-s − 2.94·37-s + ⋯
L(s)  = 1  − 0.239i·3-s + 1.32i·5-s + 0.942·9-s + 0.432·11-s − 1.17i·13-s + 0.317·15-s − 0.221i·17-s + 0.220i·19-s + 1.20·23-s − 0.758·25-s − 0.464i·27-s + 0.121·29-s + 1.57i·31-s − 0.103i·33-s − 0.0794·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.755 - 0.654i$
Analytic conductor: \(21.3624\)
Root analytic conductor: \(4.62195\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1),\ 0.755 - 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.074218423\)
\(L(\frac12)\) \(\approx\) \(2.074218423\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + 0.717iT - 9T^{2} \)
5 \( 1 - 6.63iT - 25T^{2} \)
11 \( 1 - 4.75T + 121T^{2} \)
13 \( 1 + 15.2iT - 169T^{2} \)
17 \( 1 + 3.76iT - 289T^{2} \)
19 \( 1 - 4.18iT - 361T^{2} \)
23 \( 1 - 27.7T + 529T^{2} \)
29 \( 1 - 3.51T + 841T^{2} \)
31 \( 1 - 48.8iT - 961T^{2} \)
37 \( 1 + 2.94T + 1.36e3T^{2} \)
41 \( 1 - 27.9iT - 1.68e3T^{2} \)
43 \( 1 - 10.4T + 1.84e3T^{2} \)
47 \( 1 - 52.6iT - 2.20e3T^{2} \)
53 \( 1 - 55.9T + 2.80e3T^{2} \)
59 \( 1 + 38.7iT - 3.48e3T^{2} \)
61 \( 1 - 90.5iT - 3.72e3T^{2} \)
67 \( 1 - 34.6T + 4.48e3T^{2} \)
71 \( 1 + 36.4T + 5.04e3T^{2} \)
73 \( 1 - 52.6iT - 5.32e3T^{2} \)
79 \( 1 - 33.7T + 6.24e3T^{2} \)
83 \( 1 + 127. iT - 6.88e3T^{2} \)
89 \( 1 - 50.3iT - 7.92e3T^{2} \)
97 \( 1 + 101. iT - 9.40e3T^{2} \)
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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

−10.39491161006755256598460981205, −9.502115805819693403255424210338, −8.380861650680170934371424039780, −7.31690462574291329053627752769, −6.92650545377307810925658768510, −5.96346727542862109996115529463, −4.77602409303064675170529624137, −3.49719150003068708026566182833, −2.69930866769030924266879354559, −1.17708446343672629104467623267, 0.878648740007009799645836980992, 1.99410680223310063621471521614, 3.84122691442165994928757844677, 4.50754044123306784611074641329, 5.32405601070657844209128040696, 6.56046244405362461100028226537, 7.38988779466548843614720465688, 8.532743716728561220887450122781, 9.191815413371649247888658140780, 9.728512969424664185067414100611

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