Properties

Label 2-28e2-16.13-c1-0-23
Degree $2$
Conductor $784$
Sign $0.557 + 0.829i$
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  + (−1.27 + 0.605i)2-s + (−1.39 − 1.39i)3-s + (1.26 − 1.54i)4-s + (−2.16 + 2.16i)5-s + (2.62 + 0.935i)6-s + (−0.681 + 2.74i)8-s + 0.871i·9-s + (1.45 − 4.07i)10-s + (−3.09 + 3.09i)11-s + (−3.91 + 0.391i)12-s + (−1.75 − 1.75i)13-s + 6.02·15-s + (−0.791 − 3.92i)16-s + 5.20·17-s + (−0.527 − 1.11i)18-s + (0.851 + 0.851i)19-s + ⋯
L(s)  = 1  + (−0.903 + 0.428i)2-s + (−0.803 − 0.803i)3-s + (0.633 − 0.773i)4-s + (−0.968 + 0.968i)5-s + (1.06 + 0.381i)6-s + (−0.240 + 0.970i)8-s + 0.290i·9-s + (0.460 − 1.28i)10-s + (−0.933 + 0.933i)11-s + (−1.13 + 0.112i)12-s + (−0.486 − 0.486i)13-s + 1.55·15-s + (−0.197 − 0.980i)16-s + 1.26·17-s + (−0.124 − 0.262i)18-s + (0.195 + 0.195i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.557 + 0.829i$
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: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.557 + 0.829i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.333636 - 0.177726i\)
\(L(\frac12)\) \(\approx\) \(0.333636 - 0.177726i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 - 0.605i)T \)
7 \( 1 \)
good3 \( 1 + (1.39 + 1.39i)T + 3iT^{2} \)
5 \( 1 + (2.16 - 2.16i)T - 5iT^{2} \)
11 \( 1 + (3.09 - 3.09i)T - 11iT^{2} \)
13 \( 1 + (1.75 + 1.75i)T + 13iT^{2} \)
17 \( 1 - 5.20T + 17T^{2} \)
19 \( 1 + (-0.851 - 0.851i)T + 19iT^{2} \)
23 \( 1 - 6.15iT - 23T^{2} \)
29 \( 1 + (6.24 + 6.24i)T + 29iT^{2} \)
31 \( 1 - 2.78T + 31T^{2} \)
37 \( 1 + (-4.11 + 4.11i)T - 37iT^{2} \)
41 \( 1 + 6.32iT - 41T^{2} \)
43 \( 1 + (-3.05 + 3.05i)T - 43iT^{2} \)
47 \( 1 + 3.60T + 47T^{2} \)
53 \( 1 + (-5.28 + 5.28i)T - 53iT^{2} \)
59 \( 1 + (-7.13 + 7.13i)T - 59iT^{2} \)
61 \( 1 + (1.03 + 1.03i)T + 61iT^{2} \)
67 \( 1 + (0.966 + 0.966i)T + 67iT^{2} \)
71 \( 1 + 10.0iT - 71T^{2} \)
73 \( 1 + 15.1iT - 73T^{2} \)
79 \( 1 + 6.61T + 79T^{2} \)
83 \( 1 + (-7.41 - 7.41i)T + 83iT^{2} \)
89 \( 1 + 3.26iT - 89T^{2} \)
97 \( 1 - 7.66T + 97T^{2} \)
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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

−10.17267332832652555398041820750, −9.481352896821625239479004657435, −7.80591706349733977130743737526, −7.64310041776216852776883679208, −7.04651042810679876447818546933, −5.94914774293891601179122632074, −5.24171683747922387259584335279, −3.47249720761624434005972353935, −2.09099771087115350517310177641, −0.39773857421787832328786389966, 0.869354371355108214558356467390, 2.86407167262564373905458675766, 4.06910607016425087539905423826, 4.92482618987164362841801783287, 5.88786639841079195974545611721, 7.30513954775841506993636170033, 8.111071702210639390293344559930, 8.704563881540627272661816876540, 9.758651286278014838822489323390, 10.39775172301503554996018975246

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