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Properties

Label 2-1260-28.27-c1-0-77
Degree $2$
Conductor $1260$
Sign $-0.327 + 0.944i$
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  + (1.36 − 0.366i)2-s + (1.73 − i)4-s − i·5-s + (−1.73 − 2i)7-s + (1.99 − 2i)8-s + (−0.366 − 1.36i)10-s + 3.73i·11-s − 6.46i·13-s + (−3.09 − 2.09i)14-s + (1.99 − 3.46i)16-s + 0.464i·17-s − 6·19-s + (−1 − 1.73i)20-s + (1.36 + 5.09i)22-s − 5.46i·23-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.5i)4-s − 0.447i·5-s + (−0.654 − 0.755i)7-s + (0.707 − 0.707i)8-s + (−0.115 − 0.431i)10-s + 1.12i·11-s − 1.79i·13-s + (−0.827 − 0.560i)14-s + (0.499 − 0.866i)16-s + 0.112i·17-s − 1.37·19-s + (−0.223 − 0.387i)20-s + (0.291 + 1.08i)22-s − 1.13i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.327 + 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.327 + 0.944i$
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: \((2,\ 1260,\ (\ :1/2),\ -0.327 + 0.944i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.595332202\)
\(L(\frac12)\) \(\approx\) \(2.595332202\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
3 \( 1 \)
5 \( 1 + iT \)
7 \( 1 + (1.73 + 2i)T \)
good11 \( 1 - 3.73iT - 11T^{2} \)
13 \( 1 + 6.46iT - 13T^{2} \)
17 \( 1 - 0.464iT - 17T^{2} \)
19 \( 1 + 6T + 19T^{2} \)
23 \( 1 + 5.46iT - 23T^{2} \)
29 \( 1 - 5.92T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 2.53T + 37T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 1.73T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 3.46T + 59T^{2} \)
61 \( 1 + 2.53iT - 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 0.535iT - 71T^{2} \)
73 \( 1 - 0.928iT - 73T^{2} \)
79 \( 1 - 2.66iT - 79T^{2} \)
83 \( 1 - 8.53T + 83T^{2} \)
89 \( 1 - 9.46iT - 89T^{2} \)
97 \( 1 + 7.39iT - 97T^{2} \)
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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

−9.929006098378774323657946359993, −8.504623396314461153731795593454, −7.68630907936383634181799290716, −6.70942207062398585066244545778, −6.12375857990017982652123914162, −4.91622927041094039839603390819, −4.38265057850529884408424905605, −3.30947402138291741896206331044, −2.31209634573333695917693431686, −0.76785247261405926814438718993, 1.99017824741381877206560464709, 2.96335085214019490004100720949, 3.84893615351133428777818958591, 4.80946036397823222361816509719, 5.98889347676214098991509499741, 6.39288908155171626429376229856, 7.12117902386569019126286686082, 8.345521268283386066157262931747, 8.937355180047252335105255185467, 10.01548166416559250607679353306

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