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Properties

Label 2-28e2-16.13-c1-0-26
Degree $2$
Conductor $784$
Sign $-0.669 - 0.743i$
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  + (−0.428 + 1.34i)2-s + (1.51 + 1.51i)3-s + (−1.63 − 1.15i)4-s + (0.726 − 0.726i)5-s + (−2.69 + 1.39i)6-s + (2.25 − 1.70i)8-s + 1.59i·9-s + (0.667 + 1.28i)10-s + (−0.931 + 0.931i)11-s + (−0.726 − 4.22i)12-s + (3.98 + 3.98i)13-s + 2.20·15-s + (1.33 + 3.77i)16-s − 5.62·17-s + (−2.15 − 0.683i)18-s + (3.43 + 3.43i)19-s + ⋯
L(s)  = 1  + (−0.302 + 0.953i)2-s + (0.875 + 0.875i)3-s + (−0.816 − 0.577i)4-s + (0.324 − 0.324i)5-s + (−1.09 + 0.569i)6-s + (0.797 − 0.603i)8-s + 0.531i·9-s + (0.211 + 0.407i)10-s + (−0.280 + 0.280i)11-s + (−0.209 − 1.21i)12-s + (1.10 + 1.10i)13-s + 0.568·15-s + (0.333 + 0.942i)16-s − 1.36·17-s + (−0.506 − 0.161i)18-s + (0.788 + 0.788i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.669 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.669 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.669 - 0.743i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.669 - 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.687728 + 1.54478i\)
\(L(\frac12)\) \(\approx\) \(0.687728 + 1.54478i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.428 - 1.34i)T \)
7 \( 1 \)
good3 \( 1 + (-1.51 - 1.51i)T + 3iT^{2} \)
5 \( 1 + (-0.726 + 0.726i)T - 5iT^{2} \)
11 \( 1 + (0.931 - 0.931i)T - 11iT^{2} \)
13 \( 1 + (-3.98 - 3.98i)T + 13iT^{2} \)
17 \( 1 + 5.62T + 17T^{2} \)
19 \( 1 + (-3.43 - 3.43i)T + 19iT^{2} \)
23 \( 1 - 0.572iT - 23T^{2} \)
29 \( 1 + (-5.81 - 5.81i)T + 29iT^{2} \)
31 \( 1 + 2.54T + 31T^{2} \)
37 \( 1 + (5.51 - 5.51i)T - 37iT^{2} \)
41 \( 1 + 4.12iT - 41T^{2} \)
43 \( 1 + (-0.783 + 0.783i)T - 43iT^{2} \)
47 \( 1 - 9.54T + 47T^{2} \)
53 \( 1 + (3.45 - 3.45i)T - 53iT^{2} \)
59 \( 1 + (-10.1 + 10.1i)T - 59iT^{2} \)
61 \( 1 + (6.35 + 6.35i)T + 61iT^{2} \)
67 \( 1 + (-6.03 - 6.03i)T + 67iT^{2} \)
71 \( 1 + 9.57iT - 71T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 + 8.58T + 79T^{2} \)
83 \( 1 + (2.65 + 2.65i)T + 83iT^{2} \)
89 \( 1 + 2.82iT - 89T^{2} \)
97 \( 1 + 8.98T + 97T^{2} \)
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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

−10.25697412984543380748355645315, −9.391617762012374836681911820855, −8.909887789771608375910430593292, −8.369722101770031927154092830113, −7.16989149427535126344214529931, −6.33751976375427655102125245603, −5.20637566403859226100497832339, −4.34713438407665924111478008608, −3.46136248623753125993507164185, −1.65972477136621967543372523463, 0.930777757255715092360971319477, 2.36164515489703980775141491514, 2.89128816179002811697920394388, 4.12013445742201897590095349614, 5.49110735866654762924607860174, 6.73072617607900751320934433587, 7.68543302453174767441048400071, 8.488136860277024551625810227624, 8.919735337171332380597404664286, 10.10268586237697901406608305529

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