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Properties

Label 2-3060-85.64-c1-0-2
Degree $2$
Conductor $3060$
Sign $-0.529 - 0.848i$
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  + (−1.47 − 1.68i)5-s + (0.500 + 0.500i)7-s + (−3.65 + 3.65i)11-s − 6.38i·13-s + (1.97 + 3.61i)17-s − 7.64i·19-s + (−2.47 − 2.47i)23-s + (−0.649 + 4.95i)25-s + (−0.963 − 0.963i)29-s + (6.37 + 6.37i)31-s + (0.103 − 1.58i)35-s + (−6.04 + 6.04i)37-s + (−1.16 + 1.16i)41-s + 2.36·43-s + 8.23i·47-s + ⋯
L(s)  = 1  + (−0.659 − 0.751i)5-s + (0.189 + 0.189i)7-s + (−1.10 + 1.10i)11-s − 1.76i·13-s + (0.479 + 0.877i)17-s − 1.75i·19-s + (−0.516 − 0.516i)23-s + (−0.129 + 0.991i)25-s + (−0.178 − 0.178i)29-s + (1.14 + 1.14i)31-s + (0.0174 − 0.267i)35-s + (−0.993 + 0.993i)37-s + (−0.182 + 0.182i)41-s + 0.360·43-s + 1.20i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.529 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $-0.529 - 0.848i$
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: \((2,\ 3060,\ (\ :1/2),\ -0.529 - 0.848i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3384066047\)
\(L(\frac12)\) \(\approx\) \(0.3384066047\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.47 + 1.68i)T \)
17 \( 1 + (-1.97 - 3.61i)T \)
good7 \( 1 + (-0.500 - 0.500i)T + 7iT^{2} \)
11 \( 1 + (3.65 - 3.65i)T - 11iT^{2} \)
13 \( 1 + 6.38iT - 13T^{2} \)
19 \( 1 + 7.64iT - 19T^{2} \)
23 \( 1 + (2.47 + 2.47i)T + 23iT^{2} \)
29 \( 1 + (0.963 + 0.963i)T + 29iT^{2} \)
31 \( 1 + (-6.37 - 6.37i)T + 31iT^{2} \)
37 \( 1 + (6.04 - 6.04i)T - 37iT^{2} \)
41 \( 1 + (1.16 - 1.16i)T - 41iT^{2} \)
43 \( 1 - 2.36T + 43T^{2} \)
47 \( 1 - 8.23iT - 47T^{2} \)
53 \( 1 + 4.11T + 53T^{2} \)
59 \( 1 - 0.388iT - 59T^{2} \)
61 \( 1 + (4.59 - 4.59i)T - 61iT^{2} \)
67 \( 1 + 4.45iT - 67T^{2} \)
71 \( 1 + (1.73 + 1.73i)T + 71iT^{2} \)
73 \( 1 + (10.3 - 10.3i)T - 73iT^{2} \)
79 \( 1 + (-1.31 + 1.31i)T - 79iT^{2} \)
83 \( 1 + 5.97T + 83T^{2} \)
89 \( 1 - 7.54T + 89T^{2} \)
97 \( 1 + (4.02 - 4.02i)T - 97iT^{2} \)
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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

−8.722756755489629652622905097562, −8.171451936659883953299118485828, −7.68837771595700678199236510289, −6.85317924839192604023419432645, −5.72980501848988303321126448783, −4.95397767403914577340331474567, −4.58761158295299845021587978839, −3.30772554231951022532759664125, −2.52951678837535980786301790486, −1.14962204763505867081520070015, 0.11313894015134831352434869729, 1.76464085005659939850119919661, 2.85005477802023996405439663346, 3.69942867489418719052780805211, 4.37201216981080235331211784901, 5.51875902760187981353548747061, 6.16873534890769081828205221377, 7.08557538677534835845627560802, 7.74453582568101467350758707744, 8.242782336991037597648317009248

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