Properties

Label 2-28e2-112.109-c1-0-37
Degree $2$
Conductor $784$
Sign $0.801 - 0.597i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
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  + (1.41 − 0.0428i)2-s + (−0.831 + 0.222i)3-s + (1.99 − 0.121i)4-s + (2.02 + 0.543i)5-s + (−1.16 + 0.350i)6-s + (2.81 − 0.256i)8-s + (−1.95 + 1.12i)9-s + (2.88 + 0.681i)10-s + (1.03 + 3.85i)11-s + (−1.63 + 0.545i)12-s + (0.990 + 0.990i)13-s − 1.80·15-s + (3.97 − 0.483i)16-s + (−3.07 + 5.33i)17-s + (−2.71 + 1.68i)18-s + (1.01 − 3.79i)19-s + ⋯
L(s)  = 1  + (0.999 − 0.0303i)2-s + (−0.480 + 0.128i)3-s + (0.998 − 0.0605i)4-s + (0.906 + 0.242i)5-s + (−0.476 + 0.143i)6-s + (0.995 − 0.0907i)8-s + (−0.652 + 0.376i)9-s + (0.913 + 0.215i)10-s + (0.311 + 1.16i)11-s + (−0.471 + 0.157i)12-s + (0.274 + 0.274i)13-s − 0.466·15-s + (0.992 − 0.120i)16-s + (−0.746 + 1.29i)17-s + (−0.640 + 0.396i)18-s + (0.233 − 0.870i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.801 - 0.597i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.801 - 0.597i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.70033 + 0.895873i\)
\(L(\frac12)\) \(\approx\) \(2.70033 + 0.895873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.0428i)T \)
7 \( 1 \)
good3 \( 1 + (0.831 - 0.222i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-2.02 - 0.543i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.03 - 3.85i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.990 - 0.990i)T + 13iT^{2} \)
17 \( 1 + (3.07 - 5.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.01 + 3.79i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-5.91 + 3.41i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.83 + 3.83i)T + 29iT^{2} \)
31 \( 1 + (-2.05 + 3.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.0740 - 0.0198i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.68iT - 41T^{2} \)
43 \( 1 + (-0.713 + 0.713i)T - 43iT^{2} \)
47 \( 1 + (1.95 + 3.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.89 - 7.06i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.851 - 3.17i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.37 + 8.84i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.49 - 0.401i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.86iT - 71T^{2} \)
73 \( 1 + (8.95 + 5.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.33 + 5.77i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.2 + 10.2i)T + 83iT^{2} \)
89 \( 1 + (-1.16 + 0.671i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 18.7T + 97T^{2} \)
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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

−10.59501038682681174505665401003, −9.807446899014318020365867554645, −8.733572073537646564131932991938, −7.48549055251209385852865910763, −6.45703405651050710658338326768, −6.04414393450594704854691878432, −4.94764535465009358124020704354, −4.25371089665544351918650337489, −2.73370934350807829644471397951, −1.83829861513310869094827543482, 1.23256098142907119881852277505, 2.77332533472902042200166942938, 3.67496550241217254908099649903, 5.25435529979963890927718542538, 5.54714927597487174835652925375, 6.42861941231003086474260346615, 7.24068663520810317525346108650, 8.585202637998094711108350561192, 9.335902718560845793211526027559, 10.50367814999515162933794526162

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