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Properties

Label 2-28e2-28.27-c5-0-7
Degree $2$
Conductor $784$
Sign $-0.912 + 0.409i$
Analytic cond. $125.740$
Root an. cond. $11.2134$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 22.3·3-s + 90.0i·5-s + 255.·9-s + 729. i·11-s − 874. i·13-s + 2.01e3i·15-s − 1.44e3i·17-s − 2.44e3·19-s + 629. i·23-s − 4.98e3·25-s + 283.·27-s − 5.88e3·29-s − 6.14e3·31-s + 1.62e4i·33-s + 2.88e3·37-s + ⋯
L(s)  = 1  + 1.43·3-s + 1.61i·5-s + 1.05·9-s + 1.81i·11-s − 1.43i·13-s + 2.30i·15-s − 1.20i·17-s − 1.55·19-s + 0.248i·23-s − 1.59·25-s + 0.0747·27-s − 1.29·29-s − 1.14·31-s + 2.60i·33-s + 0.346·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.912 + 0.409i$
Analytic conductor: \(125.740\)
Root analytic conductor: \(11.2134\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :5/2),\ -0.912 + 0.409i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9834983515\)
\(L(\frac12)\) \(\approx\) \(0.9834983515\)
\(L(\frac{7}{2})\) 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 \( 1 - 22.3T + 243T^{2} \)
5 \( 1 - 90.0iT - 3.12e3T^{2} \)
11 \( 1 - 729. iT - 1.61e5T^{2} \)
13 \( 1 + 874. iT - 3.71e5T^{2} \)
17 \( 1 + 1.44e3iT - 1.41e6T^{2} \)
19 \( 1 + 2.44e3T + 2.47e6T^{2} \)
23 \( 1 - 629. iT - 6.43e6T^{2} \)
29 \( 1 + 5.88e3T + 2.05e7T^{2} \)
31 \( 1 + 6.14e3T + 2.86e7T^{2} \)
37 \( 1 - 2.88e3T + 6.93e7T^{2} \)
41 \( 1 - 4.10e3iT - 1.15e8T^{2} \)
43 \( 1 - 2.07e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.35e4T + 2.29e8T^{2} \)
53 \( 1 + 1.20e4T + 4.18e8T^{2} \)
59 \( 1 - 3.47e4T + 7.14e8T^{2} \)
61 \( 1 - 1.20e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.81e4iT - 1.35e9T^{2} \)
71 \( 1 - 6.73e4iT - 1.80e9T^{2} \)
73 \( 1 + 6.28e3iT - 2.07e9T^{2} \)
79 \( 1 + 1.51e4iT - 3.07e9T^{2} \)
83 \( 1 + 9.03e4T + 3.93e9T^{2} \)
89 \( 1 + 1.01e5iT - 5.58e9T^{2} \)
97 \( 1 + 9.14e4iT - 8.58e9T^{2} \)
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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

−9.933801474322250611665175655499, −9.304774992177100966838255697384, −8.178652421751093170950564533243, −7.28411302361939182022982808310, −7.11901933126427500283183010747, −5.71274404094727183304037088196, −4.31203189324685449946135858247, −3.35836807136190763114581245399, −2.56172624007392236490335977800, −1.98370214722024186452406481520, 0.13754808621665897157481468809, 1.46360619283651079278565038361, 2.22895605450023314964986808046, 3.77198204497089113938567354487, 4.09610257250770462918399215490, 5.46038689868509463932227701636, 6.38250435392378878811807523703, 7.78118616397562689046617268354, 8.547341171240971085223709457380, 8.838628132069474029511879732381

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