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Properties

Label 2-1299-1299.920-c0-0-0
Degree $2$
Conductor $1299$
Sign $0.968 + 0.249i$
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 + (0.607 − 0.465i)7-s + (−0.499 + 0.866i)9-s + (0.866 − 0.5i)12-s + (0.448 − 1.67i)13-s − 16-s + (0.258 − 0.0340i)19-s + (0.707 + 0.292i)21-s + (0.965 − 0.258i)25-s − 0.999·27-s + (−0.465 − 0.607i)28-s + (0.241 + 1.83i)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 + (0.607 − 0.465i)7-s + (−0.499 + 0.866i)9-s + (0.866 − 0.5i)12-s + (0.448 − 1.67i)13-s − 16-s + (0.258 − 0.0340i)19-s + (0.707 + 0.292i)21-s + (0.965 − 0.258i)25-s − 0.999·27-s + (−0.465 − 0.607i)28-s + (0.241 + 1.83i)31-s + (0.866 + 0.499i)36-s + (−1.41 + 1.41i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1299\)    =    \(3 \cdot 433\)
Sign: $0.968 + 0.249i$
Analytic conductor: \(0.648285\)
Root analytic conductor: \(0.805161\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1299} (920, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1299,\ (\ :0),\ 0.968 + 0.249i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.324260153\)
\(L(\frac12)\) \(\approx\) \(1.324260153\)
\(L(1)\) 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 \( 1 + (-0.965 + 0.258i)T^{2} \)
7 \( 1 + (-0.607 + 0.465i)T + (0.258 - 0.965i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
13 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.258 + 0.0340i)T + (0.965 - 0.258i)T^{2} \)
23 \( 1 + (0.965 + 0.258i)T^{2} \)
29 \( 1 + (0.965 - 0.258i)T^{2} \)
31 \( 1 + (-0.241 - 1.83i)T + (-0.965 + 0.258i)T^{2} \)
37 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.758 + 1.83i)T + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + (-0.965 - 0.258i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.965 + 1.25i)T + (-0.258 + 0.965i)T^{2} \)
67 \( 1 + (-0.607 - 1.46i)T + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (0.258 + 0.965i)T^{2} \)
73 \( 1 + (0.241 - 0.0999i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (0.965 - 0.258i)T^{2} \)
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

−10.08541163839999597470378164989, −8.952693889063619407010140963047, −8.440614038737749755711334189301, −7.49801378723243638572975410296, −6.42104275798026638888276920030, −5.15638223794915532490921303090, −5.07125316840807238058835205836, −3.73624887444878233204081155548, −2.76031988943149747025611659595, −1.27300191107056388115330689822, 1.71653239492898455792107572380, 2.59602523141065944364139416234, 3.70515156509311225205681473574, 4.58619590647513959242815200834, 5.94941023168367657326757884544, 6.81843349089724561656405058328, 7.50406439278752386501770474048, 8.228523598387355670742400911524, 8.980651043430369942094142634776, 9.377569336632786486844991036967

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