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Properties

Label 2-28e2-112.109-c1-0-59
Degree $2$
Conductor $784$
Sign $0.650 - 0.759i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.650 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.650 - 0.759i$
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: \((2,\ 784,\ (\ :1/2),\ 0.650 - 0.759i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.54476 + 1.62993i\)
\(L(\frac12)\) \(\approx\) \(3.54476 + 1.62993i\)
\(L(\frac{3}{2})\) 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

−10.54509625079618846257400908887, −9.176976026941622763247348383769, −8.657627241678096531285638221691, −7.78721350694362126739585248247, −6.87390667158528236889320421236, −5.90527500251311146844399537414, −5.34942748803086893886261125598, −3.80891078472934349877237444855, −2.79498092835966804396677165291, −2.15672058439573090337766678074, 1.80147736248783874290452451150, 2.43443997519175038929363762422, 3.53083205811637818482659724715, 4.82395879848179175074461997567, 5.30773063145052642573451557330, 6.62357477665678784393818118617, 7.44686626031548509169764984209, 8.992579282340520544255543374725, 9.506882583264618151309667228314, 9.869698500575649137745938289187

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