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Properties

Label 2-28e2-112.37-c1-0-73
Degree $2$
Conductor $784$
Sign $-0.545 + 0.838i$
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  + (1.13 − 0.844i)2-s + (−1.26 − 0.338i)3-s + (0.572 − 1.91i)4-s + (3.57 − 0.958i)5-s + (−1.72 + 0.683i)6-s + (−0.968 − 2.65i)8-s + (−1.11 − 0.642i)9-s + (3.24 − 4.10i)10-s + (1.28 − 4.80i)11-s + (−1.37 + 2.22i)12-s + (−3.14 + 3.14i)13-s − 4.84·15-s + (−3.34 − 2.19i)16-s + (3.33 + 5.76i)17-s + (−1.80 + 0.211i)18-s + (−1.26 − 4.73i)19-s + ⋯
L(s)  = 1  + (0.802 − 0.597i)2-s + (−0.730 − 0.195i)3-s + (0.286 − 0.958i)4-s + (1.59 − 0.428i)5-s + (−0.702 + 0.279i)6-s + (−0.342 − 0.939i)8-s + (−0.371 − 0.214i)9-s + (1.02 − 1.29i)10-s + (0.388 − 1.44i)11-s + (−0.396 + 0.643i)12-s + (−0.871 + 0.871i)13-s − 1.25·15-s + (−0.835 − 0.548i)16-s + (0.807 + 1.39i)17-s + (−0.425 + 0.0498i)18-s + (−0.291 − 1.08i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-0.545 + 0.838i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -0.545 + 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08504 - 2.00082i\)
\(L(\frac12)\) \(\approx\) \(1.08504 - 2.00082i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.13 + 0.844i)T \)
7 \( 1 \)
good3 \( 1 + (1.26 + 0.338i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-3.57 + 0.958i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.28 + 4.80i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (3.14 - 3.14i)T - 13iT^{2} \)
17 \( 1 + (-3.33 - 5.76i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.26 + 4.73i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.83 - 1.63i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.0287 - 0.0287i)T - 29iT^{2} \)
31 \( 1 + (2.29 + 3.97i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.28 - 0.343i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.49iT - 41T^{2} \)
43 \( 1 + (0.947 + 0.947i)T + 43iT^{2} \)
47 \( 1 + (0.399 - 0.691i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.490 - 1.83i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.40 + 5.24i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.47 + 5.52i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-10.5 - 2.83i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.16iT - 71T^{2} \)
73 \( 1 + (1.13 - 0.654i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.166 - 0.287i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.9 - 10.9i)T - 83iT^{2} \)
89 \( 1 + (-5.63 - 3.25i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 8.56T + 97T^{2} \)
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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

−10.13744431107293069887688402253, −9.361610177370669246764261061052, −8.666218003271294035399788380555, −6.82270244325332588323302033862, −6.10312096374907364738997532961, −5.63401938740480875755954911635, −4.80630411700563621490607785701, −3.39083020510312832814846970732, −2.15390005089095645942744895821, −0.987436834302730902918426584034, 2.10516815174002494689819458916, 3.06334215555791479872646882507, 4.77647651531807765922093359118, 5.29289981502593317744245757685, 5.98921681809557543318917557034, 6.90931123533294124209321388378, 7.59374213331053106673916840888, 8.970107961921825926122725745562, 9.988487820822294423846562123493, 10.39244235501509641076520190710

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