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Properties

Label 2-1299-1299.293-c0-0-0
Degree $2$
Conductor $1299$
Sign $-0.0553 - 0.998i$
Analytic cond. $0.648285$
Root an. cond. $0.805161$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + i·4-s + (0.0999 − 0.758i)7-s + (−0.499 + 0.866i)9-s + (−0.866 + 0.5i)12-s + (1.67 + 0.448i)13-s − 16-s + (−0.965 + 1.25i)19-s + (0.707 − 0.292i)21-s + (−0.258 − 0.965i)25-s − 0.999·27-s + (0.758 + 0.0999i)28-s + (1.46 + 1.12i)31-s + (−0.866 − 0.499i)36-s + (−1.41 − 1.41i)37-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + i·4-s + (0.0999 − 0.758i)7-s + (−0.499 + 0.866i)9-s + (−0.866 + 0.5i)12-s + (1.67 + 0.448i)13-s − 16-s + (−0.965 + 1.25i)19-s + (0.707 − 0.292i)21-s + (−0.258 − 0.965i)25-s − 0.999·27-s + (0.758 + 0.0999i)28-s + (1.46 + 1.12i)31-s + (−0.866 − 0.499i)36-s + (−1.41 − 1.41i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0553 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0553 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1299\)    =    \(3 \cdot 433\)
Sign: $-0.0553 - 0.998i$
Analytic conductor: \(0.648285\)
Root analytic conductor: \(0.805161\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1299} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1299,\ (\ :0),\ -0.0553 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.267481367\)
\(L(\frac12)\) \(\approx\) \(1.267481367\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
433 \( 1 - iT \)
good2 \( 1 - iT^{2} \)
5 \( 1 + (0.258 + 0.965i)T^{2} \)
7 \( 1 + (-0.0999 + 0.758i)T + (-0.965 - 0.258i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
13 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.965 - 1.25i)T + (-0.258 - 0.965i)T^{2} \)
23 \( 1 + (-0.258 + 0.965i)T^{2} \)
29 \( 1 + (-0.258 - 0.965i)T^{2} \)
31 \( 1 + (-1.46 - 1.12i)T + (0.258 + 0.965i)T^{2} \)
37 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.465 + 1.12i)T + (-0.707 + 0.707i)T^{2} \)
47 \( 1 + (0.258 - 0.965i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.258 - 0.0340i)T + (0.965 + 0.258i)T^{2} \)
67 \( 1 + (-0.0999 + 0.241i)T + (-0.707 - 0.707i)T^{2} \)
71 \( 1 + (-0.965 + 0.258i)T^{2} \)
73 \( 1 + (1.46 + 0.607i)T + (0.707 + 0.707i)T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + (-0.258 - 0.965i)T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T^{2} \)
97 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
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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

−10.34802197923412444767653540911, −8.919947516264860372633188075836, −8.565793687791217576128131840990, −7.894163629014096813696243121348, −6.88141062462671690705688012516, −5.93657861650301368697293128137, −4.56554812004962636873255573265, −3.90789248157293675602660735328, −3.36902115840974308214535691367, −1.98299085712659490101449768088, 1.13708557346026459360126825975, 2.19872277897122222116479995335, 3.24792995287254664156410033547, 4.61858988227026131826115447125, 5.76140203439937817037174764189, 6.26439293007932339844193621882, 7.02227726028698815449104201337, 8.306887953881282777614636792435, 8.678146410929691072843674177806, 9.473255793254211829320510428388

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