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Properties

Label 2-3060-85.64-c1-0-28
Degree $2$
Conductor $3060$
Sign $0.646 + 0.762i$
Analytic cond. $24.4342$
Root an. cond. $4.94309$
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  + (−2.23 − 0.0136i)5-s + (2.94 + 2.94i)7-s + (−0.611 + 0.611i)11-s − 4.50i·13-s + (0.812 − 4.04i)17-s + 1.08i·19-s + (−3.36 − 3.36i)23-s + (4.99 + 0.0609i)25-s + (2.76 + 2.76i)29-s + (−5.07 − 5.07i)31-s + (−6.54 − 6.63i)35-s + (−5.04 + 5.04i)37-s + (0.0629 − 0.0629i)41-s + 2.40·43-s − 10.9i·47-s + ⋯
L(s)  = 1  + (−0.999 − 0.00609i)5-s + (1.11 + 1.11i)7-s + (−0.184 + 0.184i)11-s − 1.24i·13-s + (0.197 − 0.980i)17-s + 0.247i·19-s + (−0.700 − 0.700i)23-s + (0.999 + 0.0121i)25-s + (0.513 + 0.513i)29-s + (−0.911 − 0.911i)31-s + (−1.10 − 1.12i)35-s + (−0.828 + 0.828i)37-s + (0.00983 − 0.00983i)41-s + 0.366·43-s − 1.59i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.646 + 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.646 + 0.762i$
Analytic conductor: \(24.4342\)
Root analytic conductor: \(4.94309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :1/2),\ 0.646 + 0.762i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.409189726\)
\(L(\frac12)\) \(\approx\) \(1.409189726\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2.23 + 0.0136i)T \)
17 \( 1 + (-0.812 + 4.04i)T \)
good7 \( 1 + (-2.94 - 2.94i)T + 7iT^{2} \)
11 \( 1 + (0.611 - 0.611i)T - 11iT^{2} \)
13 \( 1 + 4.50iT - 13T^{2} \)
19 \( 1 - 1.08iT - 19T^{2} \)
23 \( 1 + (3.36 + 3.36i)T + 23iT^{2} \)
29 \( 1 + (-2.76 - 2.76i)T + 29iT^{2} \)
31 \( 1 + (5.07 + 5.07i)T + 31iT^{2} \)
37 \( 1 + (5.04 - 5.04i)T - 37iT^{2} \)
41 \( 1 + (-0.0629 + 0.0629i)T - 41iT^{2} \)
43 \( 1 - 2.40T + 43T^{2} \)
47 \( 1 + 10.9iT - 47T^{2} \)
53 \( 1 - 5.66T + 53T^{2} \)
59 \( 1 - 9.96iT - 59T^{2} \)
61 \( 1 + (-9.25 + 9.25i)T - 61iT^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 + (3.49 + 3.49i)T + 71iT^{2} \)
73 \( 1 + (3.31 - 3.31i)T - 73iT^{2} \)
79 \( 1 + (-10.2 + 10.2i)T - 79iT^{2} \)
83 \( 1 + 1.08T + 83T^{2} \)
89 \( 1 - 15.1T + 89T^{2} \)
97 \( 1 + (-9.80 + 9.80i)T - 97iT^{2} \)
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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

−8.514987630008546001231354700943, −7.87936106773979992780734506428, −7.40863738055794373820222607784, −6.29963189751625939764102583135, −5.22828878450671946867543189902, −5.00600505952009166694015347144, −3.85389084490158123083646603536, −2.92164283231941209058644562627, −2.01373237987394981062882517661, −0.52421759057393296186967969066, 1.03955983658712099667100053126, 2.03374958899250216676913339187, 3.54841791900516324948147349193, 4.10895819667236056750976183665, 4.70650443086116597818718897524, 5.70764634490653316394210720913, 6.85789445122674296591009870200, 7.33389528109399181715917840559, 8.062617320666585618698793674459, 8.575928429306951891746962103875

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