Properties

Label 2-28e2-196.111-c1-0-15
Degree $2$
Conductor $784$
Sign $0.994 - 0.108i$
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  + (0.977 + 1.22i)3-s + (−0.470 + 0.374i)5-s + (−1.48 − 2.19i)7-s + (0.120 − 0.529i)9-s + (5.53 − 1.26i)11-s + (0.517 − 0.118i)13-s + (−0.918 − 0.209i)15-s + (0.00663 − 0.0137i)17-s + 4.54·19-s + (1.24 − 3.95i)21-s + (−0.102 − 0.212i)23-s + (−1.03 + 4.52i)25-s + (5.00 − 2.40i)27-s + (−3.07 − 1.48i)29-s + 5.84·31-s + ⋯
L(s)  = 1  + (0.564 + 0.707i)3-s + (−0.210 + 0.167i)5-s + (−0.559 − 0.828i)7-s + (0.0402 − 0.176i)9-s + (1.66 − 0.380i)11-s + (0.143 − 0.0327i)13-s + (−0.237 − 0.0541i)15-s + (0.00160 − 0.00334i)17-s + 1.04·19-s + (0.270 − 0.863i)21-s + (−0.0213 − 0.0443i)23-s + (−0.206 + 0.904i)25-s + (0.962 − 0.463i)27-s + (−0.571 − 0.275i)29-s + 1.05·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.994 - 0.108i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.994 - 0.108i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88925 + 0.102495i\)
\(L(\frac12)\) \(\approx\) \(1.88925 + 0.102495i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.48 + 2.19i)T \)
good3 \( 1 + (-0.977 - 1.22i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (0.470 - 0.374i)T + (1.11 - 4.87i)T^{2} \)
11 \( 1 + (-5.53 + 1.26i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.517 + 0.118i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 + (-0.00663 + 0.0137i)T + (-10.5 - 13.2i)T^{2} \)
19 \( 1 - 4.54T + 19T^{2} \)
23 \( 1 + (0.102 + 0.212i)T + (-14.3 + 17.9i)T^{2} \)
29 \( 1 + (3.07 + 1.48i)T + (18.0 + 22.6i)T^{2} \)
31 \( 1 - 5.84T + 31T^{2} \)
37 \( 1 + (-5.35 - 2.57i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + (-1.23 + 0.983i)T + (9.12 - 39.9i)T^{2} \)
43 \( 1 + (5.47 + 4.36i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-0.699 - 3.06i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (10.7 - 5.16i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + (-6.17 + 7.74i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (3.48 - 7.23i)T + (-38.0 - 47.6i)T^{2} \)
67 \( 1 + 5.76iT - 67T^{2} \)
71 \( 1 + (1.23 + 2.56i)T + (-44.2 + 55.5i)T^{2} \)
73 \( 1 + (-3.26 - 0.746i)T + (65.7 + 31.6i)T^{2} \)
79 \( 1 + 7.84iT - 79T^{2} \)
83 \( 1 + (3.22 - 14.1i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (4.67 + 1.06i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−10.01605949496275600524905096624, −9.525499376702668485204163857352, −8.863138714173569309673324499325, −7.75224712217506337675946258757, −6.81489658001870893528895336800, −6.07317329172842621585451404058, −4.54955129955094791769730676068, −3.71242435475901799979186812428, −3.17975716535314754289477302765, −1.12751753805087504116879371477, 1.35911628958643222415183450710, 2.54483803843678322388177898755, 3.66347775286770202953178389977, 4.84665013146029990766754840419, 6.12580409417289266557362518537, 6.80190875989151127962958865819, 7.76033426834540893278804857950, 8.576639866264590114844987488589, 9.320348894379062440231286689212, 10.00424828778204144758863012336

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