Properties

Label 2-28e2-112.109-c1-0-31
Degree $2$
Conductor $784$
Sign $0.343 - 0.939i$
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  + (−1.08 + 0.909i)2-s + (1.23 − 0.331i)3-s + (0.344 − 1.97i)4-s + (3.17 + 0.850i)5-s + (−1.03 + 1.48i)6-s + (1.41 + 2.44i)8-s + (−1.17 + 0.679i)9-s + (−4.21 + 1.96i)10-s + (0.110 + 0.410i)11-s + (−0.226 − 2.55i)12-s + (3.70 + 3.70i)13-s + 4.21·15-s + (−3.76 − 1.35i)16-s + (−1.35 + 2.34i)17-s + (0.655 − 1.80i)18-s + (−0.914 + 3.41i)19-s + ⋯
L(s)  = 1  + (−0.765 + 0.643i)2-s + (0.714 − 0.191i)3-s + (0.172 − 0.985i)4-s + (1.41 + 0.380i)5-s + (−0.423 + 0.606i)6-s + (0.501 + 0.865i)8-s + (−0.392 + 0.226i)9-s + (−1.33 + 0.621i)10-s + (0.0331 + 0.123i)11-s + (−0.0653 − 0.736i)12-s + (1.02 + 1.02i)13-s + 1.08·15-s + (−0.940 − 0.339i)16-s + (−0.328 + 0.568i)17-s + (0.154 − 0.425i)18-s + (−0.209 + 0.782i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.343 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.343 - 0.939i$
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: \((2,\ 784,\ (\ :1/2),\ 0.343 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35935 + 0.950146i\)
\(L(\frac12)\) \(\approx\) \(1.35935 + 0.950146i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.08 - 0.909i)T \)
7 \( 1 \)
good3 \( 1 + (-1.23 + 0.331i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-3.17 - 0.850i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.110 - 0.410i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.70 - 3.70i)T + 13iT^{2} \)
17 \( 1 + (1.35 - 2.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.914 - 3.41i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (4.36 - 2.51i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.464 - 0.464i)T + 29iT^{2} \)
31 \( 1 + (-3.87 + 6.71i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.24 - 1.40i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 10.5iT - 41T^{2} \)
43 \( 1 + (-8.35 + 8.35i)T - 43iT^{2} \)
47 \( 1 + (5.11 + 8.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.30 - 12.3i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.63 - 9.82i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.84 + 6.86i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (0.243 - 0.0652i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 6.38iT - 71T^{2} \)
73 \( 1 + (1.22 + 0.706i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.14 + 7.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.84 + 2.84i)T + 83iT^{2} \)
89 \( 1 + (0.471 - 0.272i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.67T + 97T^{2} \)
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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

−10.24656322374628665793020449371, −9.384761019930866868111963279892, −8.820694955321024433634987638022, −8.039039958155181958195513141552, −7.04107803076408959807079401863, −6.01719474434452358251152351569, −5.74337850318357909652820686179, −4.04811008732847845043926124721, −2.33109706622814464984677321089, −1.72352529241067662499843084299, 1.05335375102346514557054015539, 2.44311301634348730546919875803, 3.11152331694991194577469168817, 4.48677706406490022340897321366, 5.84141312865848282546789352045, 6.63433681786243611412210505878, 8.178242700163182989238171225522, 8.475970492834226121403402648961, 9.510227651490483522057215028068, 9.765621984674021987669394514629

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