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Properties

Label 2-28e2-16.5-c1-0-1
Degree $2$
Conductor $784$
Sign $-0.878 + 0.478i$
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  + (0.312 + 1.37i)2-s + (0.599 − 0.599i)3-s + (−1.80 + 0.862i)4-s + (−0.974 − 0.974i)5-s + (1.01 + 0.640i)6-s + (−1.75 − 2.21i)8-s + 2.28i·9-s + (1.04 − 1.64i)10-s + (−1.72 − 1.72i)11-s + (−0.565 + 1.59i)12-s + (−1.90 + 1.90i)13-s − 1.16·15-s + (2.51 − 3.11i)16-s − 6.71·17-s + (−3.14 + 0.712i)18-s + (−2.94 + 2.94i)19-s + ⋯
L(s)  = 1  + (0.220 + 0.975i)2-s + (0.346 − 0.346i)3-s + (−0.902 + 0.431i)4-s + (−0.436 − 0.436i)5-s + (0.414 + 0.261i)6-s + (−0.619 − 0.784i)8-s + 0.760i·9-s + (0.328 − 0.521i)10-s + (−0.519 − 0.519i)11-s + (−0.163 + 0.461i)12-s + (−0.528 + 0.528i)13-s − 0.302·15-s + (0.628 − 0.777i)16-s − 1.62·17-s + (−0.741 + 0.167i)18-s + (−0.676 + 0.676i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.878 + 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.878 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.878 + 0.478i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.878 + 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0834129 - 0.327660i\)
\(L(\frac12)\) \(\approx\) \(0.0834129 - 0.327660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.312 - 1.37i)T \)
7 \( 1 \)
good3 \( 1 + (-0.599 + 0.599i)T - 3iT^{2} \)
5 \( 1 + (0.974 + 0.974i)T + 5iT^{2} \)
11 \( 1 + (1.72 + 1.72i)T + 11iT^{2} \)
13 \( 1 + (1.90 - 1.90i)T - 13iT^{2} \)
17 \( 1 + 6.71T + 17T^{2} \)
19 \( 1 + (2.94 - 2.94i)T - 19iT^{2} \)
23 \( 1 - 5.29iT - 23T^{2} \)
29 \( 1 + (3.03 - 3.03i)T - 29iT^{2} \)
31 \( 1 + 1.19T + 31T^{2} \)
37 \( 1 + (2.25 + 2.25i)T + 37iT^{2} \)
41 \( 1 - 3.94iT - 41T^{2} \)
43 \( 1 + (7.02 + 7.02i)T + 43iT^{2} \)
47 \( 1 - 3.06T + 47T^{2} \)
53 \( 1 + (3.01 + 3.01i)T + 53iT^{2} \)
59 \( 1 + (4.96 + 4.96i)T + 59iT^{2} \)
61 \( 1 + (-9.69 + 9.69i)T - 61iT^{2} \)
67 \( 1 + (-3.55 + 3.55i)T - 67iT^{2} \)
71 \( 1 - 11.5iT - 71T^{2} \)
73 \( 1 - 10.3iT - 73T^{2} \)
79 \( 1 - 4.06T + 79T^{2} \)
83 \( 1 + (9.17 - 9.17i)T - 83iT^{2} \)
89 \( 1 + 16.9iT - 89T^{2} \)
97 \( 1 - 2.51T + 97T^{2} \)
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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

−10.78573610043747534265513512298, −9.652135907949826355211440638582, −8.633819259552334552591949967243, −8.257359040592493464614963902383, −7.35452748297515530079152850754, −6.61152633150916527688836856285, −5.42213650378996172434970549213, −4.66097800265435314358373224449, −3.65449099296854408133721890620, −2.13548155843500077698967116396, 0.14137974125465279647032442150, 2.24198287569163153590351038574, 3.08969206648217187763780104682, 4.18416631124785914545181232599, 4.84421065986918524489893538778, 6.21993351055299937393914572055, 7.22208664614762973873169228443, 8.443767491687994805498302228283, 9.091710265010818755072673901593, 9.949603167296610499220462412227

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