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Properties

Label 2-28e2-112.93-c1-0-72
Degree $2$
Conductor $784$
Sign $-0.960 - 0.277i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.960 - 0.277i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.960 - 0.277i$
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: \((2,\ 784,\ (\ :1/2),\ -0.960 - 0.277i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.293771 + 2.07723i\)
\(L(\frac12)\) \(\approx\) \(0.293771 + 2.07723i\)
\(L(\frac{3}{2})\) 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

−9.605362235267450896592587039225, −9.084824976279730316994252066968, −8.304761763617554260314488794047, −7.14423140921181179257025712938, −6.29393072386365906707732255000, −4.85411991376919976884003009856, −4.57539566331448866579228292588, −3.15153631442439256836186224895, −1.68256855275751076542452514623, −0.929415156552169627321514230918, 2.74976364741309759464002252110, 3.74647335514374867197366506034, 4.04570729363155563033952131692, 5.54570948257243052719009613194, 6.43985361882366907523477472472, 7.10356252188143408658574471383, 8.126408191697991347439987912217, 8.880739951863889865675687907220, 10.00729905742811316026792497928, 10.78752953353982677831563827310

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