Properties

Label 2-28e2-7.4-c1-0-13
Degree $2$
Conductor $784$
Sign $-0.198 + 0.980i$
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  + (−0.707 + 1.22i)3-s + (−1.41 − 2.44i)5-s + (0.500 + 0.866i)9-s + (−1 + 1.73i)11-s + 4·15-s + (0.707 − 1.22i)17-s + (−3.53 − 6.12i)19-s + (−2 − 3.46i)23-s + (−1.49 + 2.59i)25-s − 5.65·27-s + 2·29-s + (4.24 − 7.34i)31-s + (−1.41 − 2.44i)33-s + (−5 − 8.66i)37-s − 9.89·41-s + ⋯
L(s)  = 1  + (−0.408 + 0.707i)3-s + (−0.632 − 1.09i)5-s + (0.166 + 0.288i)9-s + (−0.301 + 0.522i)11-s + 1.03·15-s + (0.171 − 0.297i)17-s + (−0.811 − 1.40i)19-s + (−0.417 − 0.722i)23-s + (−0.299 + 0.519i)25-s − 1.08·27-s + 0.371·29-s + (0.762 − 1.31i)31-s + (−0.246 − 0.426i)33-s + (−0.821 − 1.42i)37-s − 1.54·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.408838 - 0.499712i\)
\(L(\frac12)\) \(\approx\) \(0.408838 - 0.499712i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3 \( 1 + (0.707 - 1.22i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.41 + 2.44i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.53 + 6.12i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-4.24 + 7.34i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 9.89T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (-1.41 - 2.44i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.41 + 2.44i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-0.707 + 1.22i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.89T + 83T^{2} \)
89 \( 1 + (-3.53 - 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 9.89T + 97T^{2} \)
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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

−10.09091730886074922195631696420, −9.232028606466782791395954632172, −8.414714116078745818429737258733, −7.62074525402199612727827071418, −6.52300640356107246176646557665, −5.17782408666456329625649290108, −4.70530017664041149978820721322, −3.94012971856354852972276658464, −2.24897778089507605317827534856, −0.34433996568311078907313493030, 1.54867837683605803585089452534, 3.12069013109595285993598872919, 3.89662370995756469430218815986, 5.41331540450567820387993550142, 6.45247403005873939319753723120, 6.88151375599204877770571602050, 7.903588485906189803377820176041, 8.526911079384064869819526624200, 10.05602168515952330380671492251, 10.46752144598612065416726544177

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