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Properties

Label 2-3060-15.2-c1-0-20
Degree $2$
Conductor $3060$
Sign $0.409 + 0.912i$
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 + 0.0844i)5-s + (2.49 + 2.49i)7-s − 1.52i·11-s + (−2.35 + 2.35i)13-s + (−0.707 + 0.707i)17-s − 7.97i·19-s + (−4.11 − 4.11i)23-s + (4.98 − 0.377i)25-s − 2.94·29-s + 0.221·31-s + (−5.79 − 5.37i)35-s + (0.703 + 0.703i)37-s − 1.32i·41-s + (−0.264 + 0.264i)43-s + (−1.52 + 1.52i)47-s + ⋯
L(s)  = 1  + (−0.999 + 0.0377i)5-s + (0.944 + 0.944i)7-s − 0.459i·11-s + (−0.653 + 0.653i)13-s + (−0.171 + 0.171i)17-s − 1.82i·19-s + (−0.857 − 0.857i)23-s + (0.997 − 0.0754i)25-s − 0.546·29-s + 0.0397·31-s + (−0.979 − 0.908i)35-s + (0.115 + 0.115i)37-s − 0.206i·41-s + (−0.0403 + 0.0403i)43-s + (−0.222 + 0.222i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3060\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 17\)
Sign: $0.409 + 0.912i$
Analytic conductor: \(24.4342\)
Root analytic conductor: \(4.94309\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3060} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3060,\ (\ :1/2),\ 0.409 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.128016676\)
\(L(\frac12)\) \(\approx\) \(1.128016676\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 - 0.0844i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (-2.49 - 2.49i)T + 7iT^{2} \)
11 \( 1 + 1.52iT - 11T^{2} \)
13 \( 1 + (2.35 - 2.35i)T - 13iT^{2} \)
19 \( 1 + 7.97iT - 19T^{2} \)
23 \( 1 + (4.11 + 4.11i)T + 23iT^{2} \)
29 \( 1 + 2.94T + 29T^{2} \)
31 \( 1 - 0.221T + 31T^{2} \)
37 \( 1 + (-0.703 - 0.703i)T + 37iT^{2} \)
41 \( 1 + 1.32iT - 41T^{2} \)
43 \( 1 + (0.264 - 0.264i)T - 43iT^{2} \)
47 \( 1 + (1.52 - 1.52i)T - 47iT^{2} \)
53 \( 1 + (4.36 + 4.36i)T + 53iT^{2} \)
59 \( 1 - 11.7T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 + (-6.52 - 6.52i)T + 67iT^{2} \)
71 \( 1 + 9.43iT - 71T^{2} \)
73 \( 1 + (2.07 - 2.07i)T - 73iT^{2} \)
79 \( 1 + 11.1iT - 79T^{2} \)
83 \( 1 + (6.69 + 6.69i)T + 83iT^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (-0.913 - 0.913i)T + 97iT^{2} \)
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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

−8.604411361682375400511513408628, −7.935941925585697861411473362815, −7.11804949984360916439157663319, −6.42902603443591807631859933281, −5.30002337197671899009765730411, −4.72588272264824180165320455616, −3.96358994989271974480449876728, −2.78451215346736797380570099869, −2.02161402763749878793112646460, −0.40956969701068679014085943956, 1.03518888598312020498272354515, 2.15251105512429993879743249657, 3.58480865712676160302478995653, 4.03076125244280514105509274552, 4.91332943486607880481717831680, 5.64731619127523596582396899866, 6.84714739290491101433968759907, 7.61778230812147419213135625306, 7.87341387006026656435739316263, 8.560116634059398900340364365030

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