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Properties

Label 2-560-20.3-c1-0-14
Degree $2$
Conductor $560$
Sign $-0.336 + 0.941i$
Analytic cond. $4.47162$
Root an. cond. $2.11462$
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  + (1.77 − 1.77i)3-s + (−2.23 − 0.115i)5-s + (0.707 + 0.707i)7-s − 3.32i·9-s − 3.79i·11-s + (−3.76 − 3.76i)13-s + (−4.17 + 3.76i)15-s + (4.22 − 4.22i)17-s − 2.26·19-s + 2.51·21-s + (0.265 − 0.265i)23-s + (4.97 + 0.515i)25-s + (−0.580 − 0.580i)27-s − 1.40i·29-s − 0.0693i·31-s + ⋯
L(s)  = 1  + (1.02 − 1.02i)3-s + (−0.998 − 0.0516i)5-s + (0.267 + 0.267i)7-s − 1.10i·9-s − 1.14i·11-s + (−1.04 − 1.04i)13-s + (−1.07 + 0.972i)15-s + (1.02 − 1.02i)17-s − 0.519·19-s + 0.548·21-s + (0.0553 − 0.0553i)23-s + (0.994 + 0.103i)25-s + (−0.111 − 0.111i)27-s − 0.261i·29-s − 0.0124i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.336 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(560\)    =    \(2^{4} \cdot 5 \cdot 7\)
Sign: $-0.336 + 0.941i$
Analytic conductor: \(4.47162\)
Root analytic conductor: \(2.11462\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{560} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 560,\ (\ :1/2),\ -0.336 + 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908926 - 1.29034i\)
\(L(\frac12)\) \(\approx\) \(0.908926 - 1.29034i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.23 + 0.115i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-1.77 + 1.77i)T - 3iT^{2} \)
11 \( 1 + 3.79iT - 11T^{2} \)
13 \( 1 + (3.76 + 3.76i)T + 13iT^{2} \)
17 \( 1 + (-4.22 + 4.22i)T - 17iT^{2} \)
19 \( 1 + 2.26T + 19T^{2} \)
23 \( 1 + (-0.265 + 0.265i)T - 23iT^{2} \)
29 \( 1 + 1.40iT - 29T^{2} \)
31 \( 1 + 0.0693iT - 31T^{2} \)
37 \( 1 + (-2.58 + 2.58i)T - 37iT^{2} \)
41 \( 1 - 4.82T + 41T^{2} \)
43 \( 1 + (8.35 - 8.35i)T - 43iT^{2} \)
47 \( 1 + (-6.39 - 6.39i)T + 47iT^{2} \)
53 \( 1 + (-5.32 - 5.32i)T + 53iT^{2} \)
59 \( 1 + 3.71T + 59T^{2} \)
61 \( 1 + 12.2T + 61T^{2} \)
67 \( 1 + (-8.67 - 8.67i)T + 67iT^{2} \)
71 \( 1 - 3.80iT - 71T^{2} \)
73 \( 1 + (7.40 + 7.40i)T + 73iT^{2} \)
79 \( 1 - 14.0T + 79T^{2} \)
83 \( 1 + (-9.62 + 9.62i)T - 83iT^{2} \)
89 \( 1 + 14.3iT - 89T^{2} \)
97 \( 1 + (7.58 - 7.58i)T - 97iT^{2} \)
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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

−10.57946914963936993240027676533, −9.315310203128636794753076093144, −8.465262475243497438676095300077, −7.73222896675560728615444823799, −7.41272032853015226920798029523, −6.02569882987420979235456679957, −4.80060231883202627970779210309, −3.29044670353478344706623102695, −2.64905011089707020989780622196, −0.825304282396766784909449499146, 2.14114907066539423490025869211, 3.53622995094193704345872111939, 4.24448387907881834352339535835, 4.97348642440618351617453703961, 6.81186758885424334738871201072, 7.66072202732944935730858082826, 8.412813871189677167997942034368, 9.320771720078052455464610675790, 10.07066005499893455906208121789, 10.76441891707299227580467681705

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