Properties

Label 2-28e2-112.37-c1-0-10
Degree $2$
Conductor $784$
Sign $0.658 - 0.752i$
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  + (1.16 − 0.804i)2-s + (−1.90 − 0.509i)3-s + (0.707 − 1.87i)4-s + (−2.95 + 0.792i)5-s + (−2.62 + 0.935i)6-s + (−0.681 − 2.74i)8-s + (0.754 + 0.435i)9-s + (−2.80 + 3.29i)10-s + (−1.13 + 4.22i)11-s + (−2.29 + 3.19i)12-s + (1.75 − 1.75i)13-s + 6.02·15-s + (−2.99 − 2.64i)16-s + (2.60 + 4.50i)17-s + (1.22 − 0.0998i)18-s + (−0.311 − 1.16i)19-s + ⋯
L(s)  = 1  + (0.822 − 0.568i)2-s + (−1.09 − 0.294i)3-s + (0.353 − 0.935i)4-s + (−1.32 + 0.354i)5-s + (−1.06 + 0.381i)6-s + (−0.240 − 0.970i)8-s + (0.251 + 0.145i)9-s + (−0.886 + 1.04i)10-s + (−0.341 + 1.27i)11-s + (−0.662 + 0.922i)12-s + (0.486 − 0.486i)13-s + 1.55·15-s + (−0.749 − 0.661i)16-s + (0.631 + 1.09i)17-s + (0.289 − 0.0235i)18-s + (−0.0715 − 0.266i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.658 - 0.752i$
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: \((2,\ 784,\ (\ :1/2),\ 0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.658159 + 0.298732i\)
\(L(\frac12)\) \(\approx\) \(0.658159 + 0.298732i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.16 + 0.804i)T \)
7 \( 1 \)
good3 \( 1 + (1.90 + 0.509i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (2.95 - 0.792i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (1.13 - 4.22i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.75 + 1.75i)T - 13iT^{2} \)
17 \( 1 + (-2.60 - 4.50i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.311 + 1.16i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.33 - 3.07i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.24 - 6.24i)T - 29iT^{2} \)
31 \( 1 + (-1.39 - 2.40i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.61 - 1.50i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.32iT - 41T^{2} \)
43 \( 1 + (-3.05 - 3.05i)T + 43iT^{2} \)
47 \( 1 + (1.80 - 3.12i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.93 + 7.21i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.61 - 9.74i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.380 - 1.42i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.32 - 0.353i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + (13.1 - 7.58i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.30 + 5.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.41 - 7.41i)T - 83iT^{2} \)
89 \( 1 + (-2.82 - 1.63i)T + (44.5 + 77.0i)T^{2} \)
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

−10.83566705523772802349599771096, −10.06024512350841836499001000555, −8.710798589716291071720562508495, −7.35991400327190193431774160595, −6.95240751539720817973760315821, −5.75290781166671610473781694643, −5.06001653561192615511334346383, −3.98625603123490881836566988300, −3.11707374997346694710414676323, −1.35933181364981920025321918909, 0.34236440799801147307202050028, 3.03566478663269881126653937797, 4.00288010430297595584957886800, 4.87083090800784726998461845308, 5.62976098154828385227430662939, 6.44976963291604082101164607402, 7.52647145293925016436232082839, 8.206451437640109364146607490893, 9.080152259274437519833698993664, 10.66393696136331590937372815041

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