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Properties

Label 2-1260-7.2-c1-0-9
Degree $2$
Conductor $1260$
Sign $-0.0725 + 0.997i$
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  + (−0.5 + 0.866i)5-s + (−2.62 − 0.358i)7-s + (2.12 + 3.67i)11-s − 5.24·13-s + (−2.12 − 3.67i)17-s + (3.5 − 6.06i)19-s + (2.12 − 3.67i)23-s + (−0.499 − 0.866i)25-s + 10.2·29-s + (−3.74 − 6.48i)31-s + (1.62 − 2.09i)35-s + (2.62 − 4.54i)37-s − 4.24·41-s − 5.24·43-s + (−3 + 5.19i)47-s + ⋯
L(s)  = 1  + (−0.223 + 0.387i)5-s + (−0.990 − 0.135i)7-s + (0.639 + 1.10i)11-s − 1.45·13-s + (−0.514 − 0.891i)17-s + (0.802 − 1.39i)19-s + (0.442 − 0.766i)23-s + (−0.0999 − 0.173i)25-s + 1.90·29-s + (−0.672 − 1.16i)31-s + (0.274 − 0.353i)35-s + (0.430 − 0.746i)37-s − 0.662·41-s − 0.799·43-s + (−0.437 + 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0725 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0725 + 0.997i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (541, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.0725 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8092055898\)
\(L(\frac12)\) \(\approx\) \(0.8092055898\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (2.62 + 0.358i)T \)
good11 \( 1 + (-2.12 - 3.67i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.24T + 13T^{2} \)
17 \( 1 + (2.12 + 3.67i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.12 + 3.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.2T + 29T^{2} \)
31 \( 1 + (3.74 + 6.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.62 + 4.54i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + 5.24T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.24 + 7.34i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.878 + 1.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.24 + 10.8i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.62 + 2.80i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (0.378 + 0.655i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.5 - 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 1.75T + 83T^{2} \)
89 \( 1 + (0.878 - 1.52i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 16.4T + 97T^{2} \)
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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

−9.712108391567951289046495325539, −8.872918170420622993740798362849, −7.55373789502975788528891420722, −6.95449197521281366842934012007, −6.51171990006710673472046886563, −5.00045797768078099113121897621, −4.43830908923356832439408447243, −3.08693055860762224970762076719, −2.36515224555647219137289890352, −0.35467799964582136721994288193, 1.32392642536948699133660662833, 2.95195106985156711257697974581, 3.66265159330290032330311514947, 4.82698715394563202024662097146, 5.78513153580815635656993120017, 6.55257730381509734267040921309, 7.42334371765893658820374234742, 8.440278739315355449864739241918, 9.006244500929728942014475435323, 9.978780601461906068843550701128

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