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Properties

Label 2-28e2-112.53-c1-0-73
Degree $2$
Conductor $784$
Sign $0.369 - 0.929i$
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  + (−0.504 − 1.32i)2-s + (−0.589 − 2.20i)3-s + (−1.49 + 1.33i)4-s + (0.622 − 2.32i)5-s + (−2.60 + 1.88i)6-s + (2.51 + 1.29i)8-s + (−1.89 + 1.09i)9-s + (−3.38 + 0.348i)10-s + (0.0284 − 0.00762i)11-s + (3.81 + 2.49i)12-s + (−4.38 + 4.38i)13-s − 5.47·15-s + (0.450 − 3.97i)16-s + (−1.36 + 2.35i)17-s + (2.40 + 1.95i)18-s + (−5.73 − 1.53i)19-s + ⋯
L(s)  = 1  + (−0.356 − 0.934i)2-s + (−0.340 − 1.27i)3-s + (−0.745 + 0.666i)4-s + (0.278 − 1.03i)5-s + (−1.06 + 0.770i)6-s + (0.888 + 0.459i)8-s + (−0.631 + 0.364i)9-s + (−1.06 + 0.110i)10-s + (0.00857 − 0.00229i)11-s + (1.09 + 0.720i)12-s + (−1.21 + 1.21i)13-s − 1.41·15-s + (0.112 − 0.993i)16-s + (−0.330 + 0.571i)17-s + (0.565 + 0.460i)18-s + (−1.31 − 0.352i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.369 - 0.929i$
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: \((2,\ 784,\ (\ :1/2),\ 0.369 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.206476 + 0.140034i\)
\(L(\frac12)\) \(\approx\) \(0.206476 + 0.140034i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.504 + 1.32i)T \)
7 \( 1 \)
good3 \( 1 + (0.589 + 2.20i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.622 + 2.32i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.0284 + 0.00762i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (4.38 - 4.38i)T - 13iT^{2} \)
17 \( 1 + (1.36 - 2.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.73 + 1.53i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.33 - 1.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.93 - 4.93i)T - 29iT^{2} \)
31 \( 1 + (-1.29 + 2.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.09 + 7.83i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.207iT - 41T^{2} \)
43 \( 1 + (0.278 + 0.278i)T + 43iT^{2} \)
47 \( 1 + (-1.91 - 3.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.22 + 0.328i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.208 + 0.0558i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.93 + 1.59i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (3.20 + 11.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.37iT - 71T^{2} \)
73 \( 1 + (-3.67 - 2.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.51 + 7.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.55 + 7.55i)T - 83iT^{2} \)
89 \( 1 + (3.03 - 1.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−9.388320136377538424433873178444, −8.989706515886777534778732600230, −7.937181137857609927008510410822, −7.19162866095747380303815360174, −6.14108148541130191579961800218, −4.92107608669902829912101719148, −4.07347199116733936453596311056, −2.20831144139494546165121963979, −1.63948757268040292189882135808, −0.14208060108000512140397887522, 2.57156917538116951315118956341, 3.99279300527182301920157033868, 4.86665240014666535027947690061, 5.72364496729900637903013394783, 6.57874071714361603422797830481, 7.50251859850578708739637300808, 8.412785595500359506182479985543, 9.513769555921249040863897215788, 10.23129527949997967136455961371, 10.39333687996717070548958528529

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