Properties

Label 2-1360-85.84-c1-0-40
Degree $2$
Conductor $1360$
Sign $-0.216 + 0.976i$
Analytic cond. $10.8596$
Root an. cond. $3.29539$
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  + 3-s + (−2 − i)5-s + 2·7-s − 2·9-s i·13-s + (−2 − i)15-s + (−1 − 4i)17-s + 5·19-s + 2·21-s − 4·23-s + (3 + 4i)25-s − 5·27-s − 9i·29-s − 5i·31-s + (−4 − 2i)35-s + ⋯
L(s)  = 1  + 0.577·3-s + (−0.894 − 0.447i)5-s + 0.755·7-s − 0.666·9-s − 0.277i·13-s + (−0.516 − 0.258i)15-s + (−0.242 − 0.970i)17-s + 1.14·19-s + 0.436·21-s − 0.834·23-s + (0.600 + 0.800i)25-s − 0.962·27-s − 1.67i·29-s − 0.898i·31-s + (−0.676 − 0.338i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1360\)    =    \(2^{4} \cdot 5 \cdot 17\)
Sign: $-0.216 + 0.976i$
Analytic conductor: \(10.8596\)
Root analytic conductor: \(3.29539\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1360} (849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1360,\ (\ :1/2),\ -0.216 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.311130245\)
\(L(\frac12)\) \(\approx\) \(1.311130245\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2 + i)T \)
17 \( 1 + (1 + 4i)T \)
good3 \( 1 - T + 3T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
19 \( 1 - 5T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 + 5iT - 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 10iT - 41T^{2} \)
43 \( 1 - 6iT - 43T^{2} \)
47 \( 1 + 7iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 - 5T + 59T^{2} \)
61 \( 1 - 5iT - 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 + 5iT - 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + 16iT - 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 - 5T + 89T^{2} \)
97 \( 1 + 7T + 97T^{2} \)
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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

−9.204499321350239169982599014922, −8.461431721359408246489103124804, −7.82232140084351518844120980245, −7.32868227586928008437788696652, −5.90357921789425646246172294165, −5.07740430291968729544691308177, −4.16454962193137831759630703757, −3.23926019641678786192941888764, −2.14435649821642831584975571915, −0.50026485977115467379362761482, 1.56839978927714775168895866166, 2.90489476939258268634211816956, 3.63617002174974162801158829612, 4.64067276973134209478790291242, 5.62885469716431337555275888599, 6.73811754420882217465280191838, 7.55928192609360918047615572455, 8.295165576441118961765660550889, 8.711920194574137199254694968870, 9.787760048323376705649460939080

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