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Properties

Label 2-28e2-112.53-c1-0-72
Degree $2$
Conductor $784$
Sign $-0.964 + 0.264i$
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  + (1.37 − 0.320i)2-s + (−0.811 − 3.02i)3-s + (1.79 − 0.882i)4-s + (0.143 − 0.537i)5-s + (−2.08 − 3.90i)6-s + (2.18 − 1.79i)8-s + (−5.90 + 3.40i)9-s + (0.0261 − 0.785i)10-s + (−3.04 + 0.814i)11-s + (−4.12 − 4.71i)12-s + (3.16 − 3.16i)13-s − 1.74·15-s + (2.44 − 3.16i)16-s + (0.490 − 0.849i)17-s + (−7.04 + 6.58i)18-s + (−7.19 − 1.92i)19-s + ⋯
L(s)  = 1  + (0.974 − 0.226i)2-s + (−0.468 − 1.74i)3-s + (0.897 − 0.441i)4-s + (0.0643 − 0.240i)5-s + (−0.851 − 1.59i)6-s + (0.774 − 0.633i)8-s + (−1.96 + 1.13i)9-s + (0.00827 − 0.248i)10-s + (−0.916 + 0.245i)11-s + (−1.19 − 1.36i)12-s + (0.877 − 0.877i)13-s − 0.449·15-s + (0.610 − 0.791i)16-s + (0.118 − 0.206i)17-s + (−1.65 + 1.55i)18-s + (−1.64 − 0.442i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.964 + 0.264i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.964 + 0.264i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.289103 - 2.14580i\)
\(L(\frac12)\) \(\approx\) \(0.289103 - 2.14580i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.37 + 0.320i)T \)
7 \( 1 \)
good3 \( 1 + (0.811 + 3.02i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.143 + 0.537i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.04 - 0.814i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.16 + 3.16i)T - 13iT^{2} \)
17 \( 1 + (-0.490 + 0.849i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.19 + 1.92i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.09 + 0.629i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.17 + 3.17i)T - 29iT^{2} \)
31 \( 1 + (2.21 - 3.83i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.236 + 0.881i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 1.21iT - 41T^{2} \)
43 \( 1 + (-0.966 - 0.966i)T + 43iT^{2} \)
47 \( 1 + (-4.98 - 8.63i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.0 + 2.95i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.47 + 0.663i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (3.53 + 0.947i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.582 + 2.17i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.934iT - 71T^{2} \)
73 \( 1 + (0.615 + 0.355i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.93 + 10.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.77 + 6.77i)T - 83iT^{2} \)
89 \( 1 + (-8.84 + 5.10i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 3.03T + 97T^{2} \)
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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

−10.51573345777870108565884059925, −8.739207173338092405991045101699, −7.88107329318914047023457190221, −7.09901374152836907219518329912, −6.28140492321215538259594606887, −5.62164610305964650129713193929, −4.68355055574644894394847116711, −3.01741719715409112397205002849, −2.09612244954654036227362101724, −0.830080036578510237400922895476, 2.52403105489327322624313850201, 3.72450271128345411582452707475, 4.28272811824418826103064221372, 5.24372997813955891607253106426, 5.97611963742726910294434233515, 6.80344853114787595869756743646, 8.311536834012581820534528037166, 8.951251713268369047827531569962, 10.37227820311434171520795733922, 10.58866510573581346855717322496

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