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Properties

Label 2-1260-60.59-c1-0-34
Degree $2$
Conductor $1260$
Sign $-0.0691 + 0.997i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
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  − 1.41i·2-s − 2.00·4-s + (−1.73 − 1.41i)5-s + 7-s + 2.82i·8-s + (−2.00 + 2.44i)10-s + 3.46·11-s + 2.44i·13-s − 1.41i·14-s + 4.00·16-s + 6.92·17-s + 2.44i·19-s + (3.46 + 2.82i)20-s − 4.89i·22-s + 1.41i·23-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + (−0.774 − 0.632i)5-s + 0.377·7-s + 1.00i·8-s + (−0.632 + 0.774i)10-s + 1.04·11-s + 0.679i·13-s − 0.377i·14-s + 1.00·16-s + 1.68·17-s + 0.561i·19-s + (0.774 + 0.632i)20-s − 1.04i·22-s + 0.294i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0691 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0691 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0691 + 0.997i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (1079, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.0691 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.428596823\)
\(L(\frac12)\) \(\approx\) \(1.428596823\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
5 \( 1 + (1.73 + 1.41i)T \)
7 \( 1 - T \)
good11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
19 \( 1 - 2.44iT - 19T^{2} \)
23 \( 1 - 1.41iT - 23T^{2} \)
29 \( 1 + 7.07iT - 29T^{2} \)
31 \( 1 + 7.34iT - 31T^{2} \)
37 \( 1 - 9.79iT - 37T^{2} \)
41 \( 1 + 2.82iT - 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + 11.3iT - 47T^{2} \)
53 \( 1 - 6.92T + 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 + 2.44iT - 73T^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 - 5.65iT - 83T^{2} \)
89 \( 1 - 5.65iT - 89T^{2} \)
97 \( 1 + 2.44iT - 97T^{2} \)
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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

−9.735051869293001713079976665155, −8.629417595495325346499996319261, −8.194744201129755546774018617556, −7.24207483772335143181179831867, −5.87497507414021054717504556016, −4.96454044222577409254578726463, −4.01999276328969179993417449405, −3.50695947202046510528366667124, −1.90869673011106777398476549036, −0.869239023628722976127759342948, 1.02113967179276266090583070717, 3.15352675575213029136399173154, 3.87015197655782717023048209341, 4.95294760888753688727276227100, 5.77114300180328126491493088507, 6.86723758353901322590105352942, 7.28094612917893430661603065937, 8.210760685825193470310290750390, 8.768994791386532733670962557734, 9.829283713981268415740609190885

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