↑

Properties

Label 2-28e2-112.109-c1-0-34
Degree $2$
Conductor $784$
Sign $0.713 - 0.700i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.688 + 1.23i)2-s + (−2.14 + 0.574i)3-s + (−1.05 − 1.70i)4-s + (3.81 + 1.02i)5-s + (0.766 − 3.04i)6-s + (2.82 − 0.126i)8-s + (1.66 − 0.960i)9-s + (−3.89 + 4.01i)10-s + (−0.326 − 1.21i)11-s + (3.22 + 3.04i)12-s + (1.21 + 1.21i)13-s − 8.77·15-s + (−1.79 + 3.57i)16-s + (3.19 − 5.53i)17-s + (0.0404 + 2.71i)18-s + (2.00 − 7.49i)19-s + ⋯
L(s)  = 1  + (−0.487 + 0.873i)2-s + (−1.23 + 0.331i)3-s + (−0.525 − 0.850i)4-s + (1.70 + 0.457i)5-s + (0.313 − 1.24i)6-s + (0.999 − 0.0446i)8-s + (0.554 − 0.320i)9-s + (−1.23 + 1.26i)10-s + (−0.0983 − 0.367i)11-s + (0.932 + 0.878i)12-s + (0.337 + 0.337i)13-s − 2.26·15-s + (−0.447 + 0.894i)16-s + (0.774 − 1.34i)17-s + (0.00952 + 0.640i)18-s + (0.460 − 1.71i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.713 - 0.700i$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ 0.713 - 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.942667 + 0.385529i\)
\(L(\frac12)\) \(\approx\) \(0.942667 + 0.385529i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.688 - 1.23i)T \)
7 \( 1 \)
good3 \( 1 + (2.14 - 0.574i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-3.81 - 1.02i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.326 + 1.21i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.21 - 1.21i)T + 13iT^{2} \)
17 \( 1 + (-3.19 + 5.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.00 + 7.49i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.99 - 1.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.44 + 4.44i)T + 29iT^{2} \)
31 \( 1 + (-3.07 + 5.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.24 - 1.40i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 3.10iT - 41T^{2} \)
43 \( 1 + (1.56 - 1.56i)T - 43iT^{2} \)
47 \( 1 + (-2.65 - 4.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.65 + 6.19i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.30 - 8.60i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.39 - 5.19i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.59 + 0.694i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 9.50iT - 71T^{2} \)
73 \( 1 + (-1.17 - 0.677i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.22 + 2.11i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.13 - 2.13i)T + 83iT^{2} \)
89 \( 1 + (-6.31 + 3.64i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.08215061851320019389205189787, −9.703310664799761150471178461979, −8.979088801338462921010614014257, −7.56783656652118792618529789983, −6.63941179382880440323909526410, −5.97844344057004359663054271473, −5.43765605954858562981194702325, −4.63903562870600720825078147735, −2.56544255540938658980570007899, −0.872286096139318647887690747290, 1.19495984384992873053069692693, 1.91626167523059083983895951988, 3.55421231194510417340615112728, 5.04051573693325151845291756405, 5.72607935781819845345294174085, 6.41355551280783436653264852447, 7.76072301629442432774339165876, 8.710484029787991752025652374583, 9.665783432120589861875669303000, 10.37675380224714509123126755964

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