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Properties

Label 2-28e2-112.109-c1-0-35
Degree $2$
Conductor $784$
Sign $0.903 + 0.429i$
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 + (−2.80 + 0.752i)3-s + (0.575 − 1.91i)4-s + (3.72 + 0.998i)5-s + (−2.55 + 3.22i)6-s + (−0.964 − 2.65i)8-s + (4.71 − 2.72i)9-s + (5.07 − 2.01i)10-s + (0.336 + 1.25i)11-s + (−0.173 + 5.80i)12-s + (1.12 + 1.12i)13-s − 11.2·15-s + (−3.33 − 2.20i)16-s + (−0.754 + 1.30i)17-s + (3.05 − 7.07i)18-s + (−0.535 + 1.99i)19-s + ⋯
L(s)  = 1  + (0.802 − 0.596i)2-s + (−1.62 + 0.434i)3-s + (0.287 − 0.957i)4-s + (1.66 + 0.446i)5-s + (−1.04 + 1.31i)6-s + (−0.340 − 0.940i)8-s + (1.57 − 0.907i)9-s + (1.60 − 0.636i)10-s + (0.101 + 0.378i)11-s + (−0.0500 + 1.67i)12-s + (0.312 + 0.312i)13-s − 2.89·15-s + (−0.834 − 0.550i)16-s + (−0.182 + 0.316i)17-s + (0.719 − 1.66i)18-s + (−0.122 + 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $0.903 + 0.429i$
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: \((2,\ 784,\ (\ :1/2),\ 0.903 + 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91635 - 0.432601i\)
\(L(\frac12)\) \(\approx\) \(1.91635 - 0.432601i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.80 - 0.752i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-3.72 - 0.998i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.336 - 1.25i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.12 - 1.12i)T + 13iT^{2} \)
17 \( 1 + (0.754 - 1.30i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.535 - 1.99i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-4.13 + 2.38i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.10 - 4.10i)T + 29iT^{2} \)
31 \( 1 + (-2.05 + 3.55i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.26 - 0.605i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.45iT - 41T^{2} \)
43 \( 1 + (-5.68 + 5.68i)T - 43iT^{2} \)
47 \( 1 + (-1.79 - 3.11i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.247 + 0.923i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.416 + 1.55i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.17 + 4.39i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-2.08 + 0.558i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 13.8iT - 71T^{2} \)
73 \( 1 + (12.5 + 7.23i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.889 - 1.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.16 - 7.16i)T + 83iT^{2} \)
89 \( 1 + (-7.32 + 4.22i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 16.2T + 97T^{2} \)
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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

−10.63091131275651651248961466380, −9.833260479611898664381785986501, −9.101471482392499965316041101310, −6.89130142228680204310704349670, −6.38810226351677061713674872831, −5.67475359000996417933267413568, −5.04324480915050315029198251956, −4.04606261658146823653390669100, −2.48165456700883068341682238330, −1.25559987456507921795576563848, 1.20875239381066728271856355822, 2.71034942881072931496834312772, 4.60180107076604114998398790760, 5.21689084318650523334030740569, 6.02285384716825043813573515160, 6.35890199872574248168639758343, 7.29588914984001897400613924077, 8.574338089115089004241513305886, 9.572871139036986189679861839023, 10.61245398909139102808285977233

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