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Properties

Label 2-28e2-112.109-c1-0-20
Degree $2$
Conductor $784$
Sign $-0.999 - 0.0274i$
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  + (0.533 + 1.30i)2-s + (−1.65 + 0.442i)3-s + (−1.43 + 1.39i)4-s + (3.54 + 0.949i)5-s + (−1.45 − 1.92i)6-s + (−2.59 − 1.12i)8-s + (−0.0695 + 0.0401i)9-s + (0.647 + 5.14i)10-s + (−0.395 − 1.47i)11-s + (1.74 − 2.93i)12-s + (2.97 + 2.97i)13-s − 6.26·15-s + (0.0933 − 3.99i)16-s + (−3.59 + 6.22i)17-s + (−0.0896 − 0.0696i)18-s + (−0.826 + 3.08i)19-s + ⋯
L(s)  = 1  + (0.377 + 0.926i)2-s + (−0.952 + 0.255i)3-s + (−0.715 + 0.698i)4-s + (1.58 + 0.424i)5-s + (−0.595 − 0.786i)6-s + (−0.917 − 0.398i)8-s + (−0.0231 + 0.0133i)9-s + (0.204 + 1.62i)10-s + (−0.119 − 0.444i)11-s + (0.503 − 0.848i)12-s + (0.826 + 0.826i)13-s − 1.61·15-s + (0.0233 − 0.999i)16-s + (−0.871 + 1.50i)17-s + (−0.0211 − 0.0164i)18-s + (−0.189 + 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.999 - 0.0274i$
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: \((2,\ 784,\ (\ :1/2),\ -0.999 - 0.0274i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0177330 + 1.29028i\)
\(L(\frac12)\) \(\approx\) \(0.0177330 + 1.29028i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.533 - 1.30i)T \)
7 \( 1 \)
good3 \( 1 + (1.65 - 0.442i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-3.54 - 0.949i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.395 + 1.47i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-2.97 - 2.97i)T + 13iT^{2} \)
17 \( 1 + (3.59 - 6.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.826 - 3.08i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.02 - 1.74i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.851 - 0.851i)T + 29iT^{2} \)
31 \( 1 + (-1.97 + 3.42i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (8.10 + 2.17i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.67iT - 41T^{2} \)
43 \( 1 + (4.25 - 4.25i)T - 43iT^{2} \)
47 \( 1 + (1.17 + 2.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.25 + 4.66i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.41 - 5.27i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.514 + 1.92i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (7.79 - 2.08i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (-2.88 - 1.66i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.90 - 13.6i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.20 - 1.20i)T + 83iT^{2} \)
89 \( 1 + (-10.9 + 6.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.08T + 97T^{2} \)
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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

−10.62588778280367173619686075000, −9.935898347130082169236566353736, −8.907704514434743283715769539617, −8.202495198705480802326381125394, −6.69342600720015574473614723499, −6.12326373598831353041206067346, −5.80828365207456403721957810059, −4.74601203471912669822538227783, −3.59153933167141209252945404634, −1.94681476070784559500243965840, 0.63613345527408041905682212227, 1.92580561946306664459549362904, 3.00836791835000271373169667919, 4.74419143354686566350005042825, 5.28064532698014995079083830481, 6.08034460693473947629235353219, 6.79973606741073504628156057508, 8.632177581828533404385524062509, 9.206634217984337731699529087093, 10.16955883267738911337241249558

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