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Properties

Label 2-28e2-112.93-c1-0-12
Degree $2$
Conductor $784$
Sign $0.411 - 0.911i$
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  + (0.867 − 1.11i)2-s + (−0.442 + 1.65i)3-s + (−0.495 − 1.93i)4-s + (0.949 + 3.54i)5-s + (1.45 + 1.92i)6-s + (−2.59 − 1.12i)8-s + (0.0695 + 0.0401i)9-s + (4.78 + 2.01i)10-s + (1.47 + 0.395i)11-s + (3.41 + 0.0398i)12-s + (−2.97 − 2.97i)13-s − 6.26·15-s + (−3.50 + 1.91i)16-s + (3.59 + 6.22i)17-s + (0.105 − 0.0428i)18-s + (−3.08 + 0.826i)19-s + ⋯
L(s)  = 1  + (0.613 − 0.789i)2-s + (−0.255 + 0.952i)3-s + (−0.247 − 0.968i)4-s + (0.424 + 1.58i)5-s + (0.595 + 0.786i)6-s + (−0.917 − 0.398i)8-s + (0.0231 + 0.0133i)9-s + (1.51 + 0.636i)10-s + (0.444 + 0.119i)11-s + (0.986 + 0.0115i)12-s + (−0.826 − 0.826i)13-s − 1.61·15-s + (−0.877 + 0.479i)16-s + (0.871 + 1.50i)17-s + (0.0247 − 0.0100i)18-s + (−0.707 + 0.189i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.411 - 0.911i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.411 - 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55752 + 1.00543i\)
\(L(\frac12)\) \(\approx\) \(1.55752 + 1.00543i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.867 + 1.11i)T \)
7 \( 1 \)
good3 \( 1 + (0.442 - 1.65i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.949 - 3.54i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.47 - 0.395i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (2.97 + 2.97i)T + 13iT^{2} \)
17 \( 1 + (-3.59 - 6.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.08 - 0.826i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.02 - 1.74i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.851 - 0.851i)T + 29iT^{2} \)
31 \( 1 + (1.97 + 3.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.17 - 8.10i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 2.67iT - 41T^{2} \)
43 \( 1 + (4.25 - 4.25i)T - 43iT^{2} \)
47 \( 1 + (-1.17 + 2.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.66 - 1.25i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.27 - 1.41i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.92 + 0.514i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.08 + 7.79i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (-2.88 + 1.66i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.90 + 13.6i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (1.20 + 1.20i)T + 83iT^{2} \)
89 \( 1 + (-10.9 - 6.31i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.08T + 97T^{2} \)
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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

−10.36430343268188712124569861672, −10.15741367623916658522891290441, −9.300654071033802327872525409945, −7.81029780959038920634009309375, −6.63469515733377848586331049180, −5.88039784713839499171075777586, −4.94868967993703417969907586134, −3.81389711430981188029301376277, −3.14134911668594920698333680545, −1.91824481197374694372744729072, 0.805239803229678138187522588629, 2.29908636703836337710636035180, 4.07338305402839559570529800076, 4.96288299906987693847501145158, 5.62540245481610396789402720398, 6.75789135910380806493616232297, 7.26412556965526461205094109262, 8.318515399892012902713607812130, 9.105187799708506589026054471260, 9.710778246177307976796635400765

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