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Properties

Label 2-2535-195.134-c0-0-6
Degree $2$
Conductor $2535$
Sign $-0.494 - 0.869i$
Analytic cond. $1.26512$
Root an. cond. $1.12477$
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  + (1.22 + 0.707i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.707 + 0.707i)5-s + (0.707 + 1.22i)6-s + (0.499 + 0.866i)9-s + (−1.36 + 0.366i)10-s + (−0.707 + 1.22i)11-s + 0.999i·12-s + (−0.965 + 0.258i)15-s + (0.499 − 0.866i)16-s + 1.41i·18-s + (−0.965 − 0.258i)20-s + (−1.73 + 0.999i)22-s − 1.00i·25-s + ⋯
L(s)  = 1  + (1.22 + 0.707i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.707 + 0.707i)5-s + (0.707 + 1.22i)6-s + (0.499 + 0.866i)9-s + (−1.36 + 0.366i)10-s + (−0.707 + 1.22i)11-s + 0.999i·12-s + (−0.965 + 0.258i)15-s + (0.499 − 0.866i)16-s + 1.41i·18-s + (−0.965 − 0.258i)20-s + (−1.73 + 0.999i)22-s − 1.00i·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.494 - 0.869i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (1499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ -0.494 - 0.869i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.609232780\)
\(L(\frac12)\) \(\approx\) \(2.609232780\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 \)
good2 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T^{2} \)
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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

−9.344579521433942848595832088125, −8.286618083364697821039308522104, −7.46662716665961755358141187915, −7.24431262609754419621975226224, −6.25227027982942626261507097584, −5.19559959088419703969145216545, −4.52774812212221856213343851977, −3.89624571914706184490732464869, −3.09154199417073983588961999013, −2.23254424608303381924714318389, 1.12437406733284238882173013038, 2.39863028226291123068114558347, 3.19285214968206362703310353980, 3.82713682089571825057722774892, 4.64588821382550641488290377509, 5.50467641083483603971422400889, 6.30919998675817710087119443986, 7.48950085835858339758988877487, 8.149852400401776385915302557638, 8.642857162101994682254449952878

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