Properties

Label 2-28e2-112.109-c1-0-3
Degree $2$
Conductor $784$
Sign $-0.999 + 0.00124i$
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  + (−0.688 + 1.23i)2-s + (2.14 − 0.574i)3-s + (−1.05 − 1.70i)4-s + (−3.81 − 1.02i)5-s + (−0.766 + 3.04i)6-s + (2.82 − 0.126i)8-s + (1.66 − 0.960i)9-s + (3.89 − 4.01i)10-s + (−0.326 − 1.21i)11-s + (−3.22 − 3.04i)12-s + (−1.21 − 1.21i)13-s − 8.77·15-s + (−1.79 + 3.57i)16-s + (−3.19 + 5.53i)17-s + (0.0404 + 2.71i)18-s + (−2.00 + 7.49i)19-s + ⋯
L(s)  = 1  + (−0.487 + 0.873i)2-s + (1.23 − 0.331i)3-s + (−0.525 − 0.850i)4-s + (−1.70 − 0.457i)5-s + (−0.313 + 1.24i)6-s + (0.999 − 0.0446i)8-s + (0.554 − 0.320i)9-s + (1.23 − 1.26i)10-s + (−0.0983 − 0.367i)11-s + (−0.932 − 0.878i)12-s + (−0.337 − 0.337i)13-s − 2.26·15-s + (−0.447 + 0.894i)16-s + (−0.774 + 1.34i)17-s + (0.00952 + 0.640i)18-s + (−0.460 + 1.71i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.999 + 0.00124i$
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: \((2,\ 784,\ (\ :1/2),\ -0.999 + 0.00124i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000153781 - 0.246239i\)
\(L(\frac12)\) \(\approx\) \(0.000153781 - 0.246239i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.688 - 1.23i)T \)
7 \( 1 \)
good3 \( 1 + (-2.14 + 0.574i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (3.81 + 1.02i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.326 + 1.21i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.21 + 1.21i)T + 13iT^{2} \)
17 \( 1 + (3.19 - 5.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.00 - 7.49i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.99 - 1.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.44 + 4.44i)T + 29iT^{2} \)
31 \( 1 + (3.07 - 5.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.24 - 1.40i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 3.10iT - 41T^{2} \)
43 \( 1 + (1.56 - 1.56i)T - 43iT^{2} \)
47 \( 1 + (2.65 + 4.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.65 + 6.19i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.30 + 8.60i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.39 + 5.19i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.59 + 0.694i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.50iT - 71T^{2} \)
73 \( 1 + (1.17 + 0.677i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.22 + 2.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.13 + 2.13i)T + 83iT^{2} \)
89 \( 1 + (6.31 - 3.64i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.8T + 97T^{2} \)
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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

−10.55497721843404406724059500092, −9.488548081362516588557898746063, −8.478866344935033259035254729658, −8.162082520086308889149491610810, −7.75932404622922314405930335399, −6.71930808233500970826493907357, −5.50949144870090161513388007766, −4.15690299279278398411266402640, −3.58337804467874122403055384355, −1.76484880506764744242889687456, 0.12125045115688964924321403446, 2.45894388155973416300718895072, 3.03635002932152058226894958775, 4.14924790244435058664368639066, 4.59532569466597218988370669311, 7.06737083137319885122586445146, 7.47683805756976385001207143001, 8.357366643877785799762815193931, 9.090214265325439770771723741476, 9.614931012973088171534618225750

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