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Properties

Label 2-28e2-16.13-c1-0-46
Degree $2$
Conductor $784$
Sign $-0.793 - 0.608i$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.36 − 0.366i)2-s + (−1.36 − 1.36i)3-s + (1.73 + i)4-s + (−2.36 + 2.36i)5-s + (1.36 + 2.36i)6-s + (−1.99 − 2i)8-s + 0.732i·9-s + (4.09 − 2.36i)10-s + (3.09 − 3.09i)11-s + (−0.999 − 3.73i)12-s + (−0.267 − 0.267i)13-s + 6.46·15-s + (1.99 + 3.46i)16-s + 0.464·17-s + (0.267 − 0.999i)18-s + (3.09 + 3.09i)19-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.788 − 0.788i)3-s + (0.866 + 0.5i)4-s + (−1.05 + 1.05i)5-s + (0.557 + 0.965i)6-s + (−0.707 − 0.707i)8-s + 0.244i·9-s + (1.29 − 0.748i)10-s + (0.934 − 0.934i)11-s + (−0.288 − 1.07i)12-s + (−0.0743 − 0.0743i)13-s + 1.66·15-s + (0.499 + 0.866i)16-s + 0.112·17-s + (0.0631 − 0.235i)18-s + (0.710 + 0.710i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.793 - 0.608i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.793 - 0.608i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
7 \( 1 \)
good3 \( 1 + (1.36 + 1.36i)T + 3iT^{2} \)
5 \( 1 + (2.36 - 2.36i)T - 5iT^{2} \)
11 \( 1 + (-3.09 + 3.09i)T - 11iT^{2} \)
13 \( 1 + (0.267 + 0.267i)T + 13iT^{2} \)
17 \( 1 - 0.464T + 17T^{2} \)
19 \( 1 + (-3.09 - 3.09i)T + 19iT^{2} \)
23 \( 1 + 2.46iT - 23T^{2} \)
29 \( 1 + (3.73 + 3.73i)T + 29iT^{2} \)
31 \( 1 + 0.267T + 31T^{2} \)
37 \( 1 + (7.83 - 7.83i)T - 37iT^{2} \)
41 \( 1 + 8.92iT - 41T^{2} \)
43 \( 1 + (0.464 - 0.464i)T - 43iT^{2} \)
47 \( 1 + 7.73T + 47T^{2} \)
53 \( 1 + (8.09 - 8.09i)T - 53iT^{2} \)
59 \( 1 + (7.29 - 7.29i)T - 59iT^{2} \)
61 \( 1 + (-0.0980 - 0.0980i)T + 61iT^{2} \)
67 \( 1 + (-5.36 - 5.36i)T + 67iT^{2} \)
71 \( 1 - 7.46iT - 71T^{2} \)
73 \( 1 + 3.19iT - 73T^{2} \)
79 \( 1 - 0.660T + 79T^{2} \)
83 \( 1 + (8.46 + 8.46i)T + 83iT^{2} \)
89 \( 1 + 5.19iT - 89T^{2} \)
97 \( 1 + 10.9T + 97T^{2} \)
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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

−9.930422391119578029265481123283, −8.825958945551441231431294848270, −7.924047185219944588683962056354, −7.20680115779047307891574116541, −6.57080556558247233605756915087, −5.82270197927015150250279661891, −3.82579587435936753097565315061, −3.07634088112798048759782239035, −1.38203089650446728811353210491, 0, 1.55425575286534673563271250846, 3.62652588910295396165505489556, 4.76706309762078478434151348488, 5.30549178648335961445073499112, 6.63273863027633782512375676789, 7.50885126110115301466016460625, 8.300926489616201349192235555156, 9.387835964224976867192975635238, 9.612785424709615844385367785056

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