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Properties

Label 2-28e2-112.53-c1-0-47
Degree $2$
Conductor $784$
Sign $-0.317 + 0.948i$
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  + (0.559 − 1.29i)2-s + (−0.224 − 0.839i)3-s + (−1.37 − 1.45i)4-s + (−0.847 + 3.16i)5-s + (−1.21 − 0.177i)6-s + (−2.65 + 0.968i)8-s + (1.94 − 1.12i)9-s + (3.63 + 2.87i)10-s + (2.87 − 0.769i)11-s + (−0.911 + 1.47i)12-s + (3.63 − 3.63i)13-s + 2.84·15-s + (−0.229 + 3.99i)16-s + (1.81 − 3.14i)17-s + (−0.369 − 3.15i)18-s + (−1.60 − 0.429i)19-s + ⋯
L(s)  = 1  + (0.395 − 0.918i)2-s + (−0.129 − 0.484i)3-s + (−0.686 − 0.727i)4-s + (−0.379 + 1.41i)5-s + (−0.496 − 0.0726i)6-s + (−0.939 + 0.342i)8-s + (0.648 − 0.374i)9-s + (1.14 + 0.908i)10-s + (0.865 − 0.231i)11-s + (−0.263 + 0.426i)12-s + (1.00 − 1.00i)13-s + 0.734·15-s + (−0.0574 + 0.998i)16-s + (0.441 − 0.763i)17-s + (−0.0870 − 0.743i)18-s + (−0.367 − 0.0985i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.317 + 0.948i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.317 + 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.972560 - 1.35180i\)
\(L(\frac12)\) \(\approx\) \(0.972560 - 1.35180i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.559 + 1.29i)T \)
7 \( 1 \)
good3 \( 1 + (0.224 + 0.839i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.847 - 3.16i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.87 + 0.769i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.63 + 3.63i)T - 13iT^{2} \)
17 \( 1 + (-1.81 + 3.14i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.60 + 0.429i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (5.33 - 3.08i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.10 + 5.10i)T - 29iT^{2} \)
31 \( 1 + (-1.00 + 1.74i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.49 + 5.57i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 3.71iT - 41T^{2} \)
43 \( 1 + (2.91 + 2.91i)T + 43iT^{2} \)
47 \( 1 + (-5.06 - 8.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.68 + 0.986i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.64 + 0.977i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.54 - 1.75i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (1.57 + 5.88i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.55iT - 71T^{2} \)
73 \( 1 + (-0.989 - 0.571i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.120 - 0.209i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.459 + 0.459i)T - 83iT^{2} \)
89 \( 1 + (3.76 - 2.17i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.80T + 97T^{2} \)
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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

−10.23531010103336011297854987840, −9.553726331854407741840635186033, −8.325830784058353768279518670576, −7.33668034620979710493859131353, −6.38715109005921436538363428982, −5.77127391755548899918229270191, −4.09298899149375767116712686672, −3.50520975201584675791737846311, −2.39919438812675828423704254369, −0.892752394078634072700796779691, 1.41038772674305009362128177627, 3.82989813256521388043584710756, 4.28703139318061908761939548065, 5.02742501510861127690676562901, 6.14495256596787900354767069504, 6.94392434163437549690033888972, 8.256402667879777503507972622087, 8.565120122859064015160475806619, 9.448665204129764572422977413084, 10.34684292281801617263475932614

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