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Properties

Label 2-28e2-112.109-c1-0-32
Degree $2$
Conductor $784$
Sign $-0.265 + 0.963i$
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  + (−0.561 − 1.29i)2-s + (−0.614 + 0.164i)3-s + (−1.37 + 1.45i)4-s + (−1.45 − 0.389i)5-s + (0.558 + 0.705i)6-s + (2.65 + 0.961i)8-s + (−2.24 + 1.29i)9-s + (0.309 + 2.10i)10-s + (0.899 + 3.35i)11-s + (0.602 − 1.12i)12-s + (−0.433 − 0.433i)13-s + 0.957·15-s + (−0.244 − 3.99i)16-s + (3.24 − 5.62i)17-s + (2.94 + 2.18i)18-s + (0.341 − 1.27i)19-s + ⋯
L(s)  = 1  + (−0.396 − 0.917i)2-s + (−0.354 + 0.0950i)3-s + (−0.685 + 0.728i)4-s + (−0.649 − 0.174i)5-s + (0.228 + 0.287i)6-s + (0.940 + 0.339i)8-s + (−0.749 + 0.432i)9-s + (0.0980 + 0.665i)10-s + (0.271 + 1.01i)11-s + (0.173 − 0.323i)12-s + (−0.120 − 0.120i)13-s + 0.247·15-s + (−0.0610 − 0.998i)16-s + (0.787 − 1.36i)17-s + (0.694 + 0.516i)18-s + (0.0783 − 0.292i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.265 + 0.963i$
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: \((2,\ 784,\ (\ :1/2),\ -0.265 + 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420519 - 0.552235i\)
\(L(\frac12)\) \(\approx\) \(0.420519 - 0.552235i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.561 + 1.29i)T \)
7 \( 1 \)
good3 \( 1 + (0.614 - 0.164i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (1.45 + 0.389i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.899 - 3.35i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.433 + 0.433i)T + 13iT^{2} \)
17 \( 1 + (-3.24 + 5.62i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.341 + 1.27i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.18 + 0.682i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.40 - 2.40i)T + 29iT^{2} \)
31 \( 1 + (-3.56 + 6.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.87 - 1.30i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.83iT - 41T^{2} \)
43 \( 1 + (-0.978 + 0.978i)T - 43iT^{2} \)
47 \( 1 + (6.31 + 10.9i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.34 + 8.76i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (1.92 + 7.18i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.57 + 5.88i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-15.4 + 4.14i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 8.41iT - 71T^{2} \)
73 \( 1 + (-0.483 - 0.279i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.458 - 0.794i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.41 - 2.41i)T + 83iT^{2} \)
89 \( 1 + (13.9 - 8.08i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.39T + 97T^{2} \)
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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

−9.904224338086764584780937186131, −9.557185011499486244250489318950, −8.289645309840652335523177702002, −7.81092158328194995154026247414, −6.72772529970088933548484868700, −5.16105743616688390424745294607, −4.59125569646731490159510367016, −3.35093785520625306032911169202, −2.27834183138458155184541657660, −0.53344308972264639390407581078, 1.05910357358524517862650982166, 3.27566884188399623119012951303, 4.27982122027263705471799041145, 5.67777725236965468636902806914, 6.06419636340553535839566406816, 7.08281529170633428544049537671, 8.088756589475751787218007883313, 8.548436142622492398696519268712, 9.523849831919153988834121043376, 10.53303824785153314657621244352

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