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Properties

Label 2-2535-195.137-c0-0-0
Degree $2$
Conductor $2535$
Sign $-0.927 + 0.374i$
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  + (−0.382 + 0.662i)2-s + (0.258 + 0.965i)3-s + (0.207 + 0.358i)4-s + (0.923 + 0.382i)5-s + (−0.739 − 0.198i)6-s − 1.08·8-s + (−0.866 + 0.499i)9-s + (−0.607 + 0.465i)10-s + (−1.78 + 0.478i)11-s + (−0.292 + 0.292i)12-s + (−0.130 + 0.991i)15-s + (0.207 − 0.358i)16-s − 0.765i·18-s + (0.0540 + 0.410i)20-s + (0.366 − 1.36i)22-s + ⋯
L(s)  = 1  + (−0.382 + 0.662i)2-s + (0.258 + 0.965i)3-s + (0.207 + 0.358i)4-s + (0.923 + 0.382i)5-s + (−0.739 − 0.198i)6-s − 1.08·8-s + (−0.866 + 0.499i)9-s + (−0.607 + 0.465i)10-s + (−1.78 + 0.478i)11-s + (−0.292 + 0.292i)12-s + (−0.130 + 0.991i)15-s + (0.207 − 0.358i)16-s − 0.765i·18-s + (0.0540 + 0.410i)20-s + (0.366 − 1.36i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2535 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2535\)    =    \(3 \cdot 5 \cdot 13^{2}\)
Sign: $-0.927 + 0.374i$
Analytic conductor: \(1.26512\)
Root analytic conductor: \(1.12477\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2535} (1502, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2535,\ (\ :0),\ -0.927 + 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9715930047\)
\(L(\frac12)\) \(\approx\) \(0.9715930047\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 \)
good2 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (1.78 - 0.478i)T + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.198 - 0.739i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + 1.84iT - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.739 - 0.198i)T + (0.866 + 0.5i)T^{2} \)
61 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.739 - 0.198i)T + (0.866 + 0.5i)T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 - 1.84iT - T^{2} \)
89 \( 1 + (-0.478 - 1.78i)T + (-0.866 + 0.5i)T^{2} \)
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

−9.655731010478300316024032114309, −8.588706364400053783969449521088, −8.144711068065558429422799862668, −7.29135004695216597441100561644, −6.52929531185057306584200199231, −5.50707123111337849631356064378, −5.15590338632358488289234800384, −3.85742470852018844529228451725, −2.79200441063569255346849651521, −2.35140920958211466432524100732, 0.63701745340125583744607216748, 1.84760900354423673796106687737, 2.48271787118400058428261647589, 3.23581145824854334202855946017, 4.99096485296589876415707386347, 5.71502554623927330021787568422, 6.20170551018207720697842644454, 7.20541854893998811734890714366, 8.037309945952326625559691077926, 8.773986820472716314854181734346

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