Properties

Label 2-1260-28.27-c1-0-2
Degree $2$
Conductor $1260$
Sign $-0.791 + 0.610i$
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  + (−1.07 + 0.921i)2-s + (0.303 − 1.97i)4-s + i·5-s + (−1.82 + 1.91i)7-s + (1.49 + 2.40i)8-s + (−0.921 − 1.07i)10-s + 6.24i·11-s − 2.40i·13-s + (0.195 − 3.73i)14-s + (−3.81 − 1.19i)16-s + 1.30i·17-s − 3.94·19-s + (1.97 + 0.303i)20-s + (−5.74 − 6.69i)22-s − 3.55i·23-s + ⋯
L(s)  = 1  + (−0.758 + 0.651i)2-s + (0.151 − 0.988i)4-s + 0.447i·5-s + (−0.690 + 0.723i)7-s + (0.528 + 0.848i)8-s + (−0.291 − 0.339i)10-s + 1.88i·11-s − 0.666i·13-s + (0.0522 − 0.998i)14-s + (−0.954 − 0.299i)16-s + 0.317i·17-s − 0.905·19-s + (0.442 + 0.0677i)20-s + (−1.22 − 1.42i)22-s − 0.741i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.791 + 0.610i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.791 + 0.610i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3429099979\)
\(L(\frac12)\) \(\approx\) \(0.3429099979\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.07 - 0.921i)T \)
3 \( 1 \)
5 \( 1 - iT \)
7 \( 1 + (1.82 - 1.91i)T \)
good11 \( 1 - 6.24iT - 11T^{2} \)
13 \( 1 + 2.40iT - 13T^{2} \)
17 \( 1 - 1.30iT - 17T^{2} \)
19 \( 1 + 3.94T + 19T^{2} \)
23 \( 1 + 3.55iT - 23T^{2} \)
29 \( 1 + 1.44T + 29T^{2} \)
31 \( 1 - 10.9T + 31T^{2} \)
37 \( 1 + 5.06T + 37T^{2} \)
41 \( 1 + 1.26iT - 41T^{2} \)
43 \( 1 - 2.19iT - 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + 0.415T + 59T^{2} \)
61 \( 1 + 12.6iT - 61T^{2} \)
67 \( 1 - 3.10iT - 67T^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + 1.64iT - 73T^{2} \)
79 \( 1 - 9.18iT - 79T^{2} \)
83 \( 1 + 7.39T + 83T^{2} \)
89 \( 1 + 11.4iT - 89T^{2} \)
97 \( 1 - 12.4iT - 97T^{2} \)
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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

−9.993874455856066630075189025208, −9.485012249976236327549064223210, −8.442701291136009190783497252108, −7.83805182545097418458558522917, −6.63634171756331439113360191830, −6.53025739266478436823345265905, −5.29058997376771331841159685614, −4.41537713394945001098033970117, −2.80811339429930452825710185865, −1.85422872506046728299602927891, 0.18900517124329574358531436916, 1.37433869576112744664532976990, 2.92063470765350886182836755447, 3.66356963159417691733972611854, 4.64238498711069521137982658047, 6.10522522861294387626976664047, 6.76335071956734248011774168349, 7.87109854752387849347291182680, 8.522375790502496869221758430019, 9.219180614046434768287312473964

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