Properties

Label 2-1560-5.4-c1-0-23
Degree $2$
Conductor $1560$
Sign $0.807 + 0.590i$
Analytic cond. $12.4566$
Root an. cond. $3.52939$
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  + i·3-s + (1.32 − 1.80i)5-s − 9-s − 4.64·11-s + i·13-s + (1.80 + 1.32i)15-s − 4.24i·17-s + 6.24·19-s + 2.24i·23-s + (−1.51 − 4.76i)25-s i·27-s + 9.21·29-s + 9.28·31-s − 4.64i·33-s − 7.28i·37-s + ⋯
L(s)  = 1  + 0.577i·3-s + (0.590 − 0.807i)5-s − 0.333·9-s − 1.39·11-s + 0.277i·13-s + (0.466 + 0.340i)15-s − 1.03i·17-s + 1.43·19-s + 0.469i·23-s + (−0.303 − 0.952i)25-s − 0.192i·27-s + 1.71·29-s + 1.66·31-s − 0.807i·33-s − 1.19i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1560\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.807 + 0.590i$
Analytic conductor: \(12.4566\)
Root analytic conductor: \(3.52939\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1560} (1249, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1560,\ (\ :1/2),\ 0.807 + 0.590i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.691165838\)
\(L(\frac12)\) \(\approx\) \(1.691165838\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
5 \( 1 + (-1.32 + 1.80i)T \)
13 \( 1 - iT \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
17 \( 1 + 4.24iT - 17T^{2} \)
19 \( 1 - 6.24T + 19T^{2} \)
23 \( 1 - 2.24iT - 23T^{2} \)
29 \( 1 - 9.21T + 29T^{2} \)
31 \( 1 - 9.28T + 31T^{2} \)
37 \( 1 + 7.28iT - 37T^{2} \)
41 \( 1 + 5.67T + 41T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 + 2.88iT - 47T^{2} \)
53 \( 1 + 9.21iT - 53T^{2} \)
59 \( 1 + 5.92T + 59T^{2} \)
61 \( 1 - 0.969T + 61T^{2} \)
67 \( 1 - 1.93iT - 67T^{2} \)
71 \( 1 - 5.60T + 71T^{2} \)
73 \( 1 + 12.5iT - 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 + 3.67iT - 83T^{2} \)
89 \( 1 - 9.67T + 89T^{2} \)
97 \( 1 - 6iT - 97T^{2} \)
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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

−9.480714188220271973593980236037, −8.635013415150993266327540199425, −7.927900575240723868640596805603, −6.97819884587705216246658747807, −5.80774251948832226749467556079, −5.11171818288867438888292609926, −4.63183174145057538038395255810, −3.23531791428257812801357020721, −2.32965637623613612762920136390, −0.73764770233626992639436083242, 1.23068546542277125054153530126, 2.64252810369303682756033193900, 3.05450190799556185236435560736, 4.64982929997061365432111977485, 5.59357451925045113239189400816, 6.30274074092437739551701187249, 7.06639634395306032651721100231, 7.964163989842803213976498035922, 8.441942384824842617661871719080, 9.778573235752966630884217695802

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