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Properties

Label 2-1260-5.4-c3-0-43
Degree $2$
Conductor $1260$
Sign $-0.735 - 0.677i$
Analytic cond. $74.3424$
Root an. cond. $8.62220$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (7.57 − 8.22i)5-s − 7i·7-s − 58.2·11-s − 70.3i·13-s + 44.7i·17-s + 26.8·19-s − 18.8i·23-s + (−10.2 − 124. i)25-s − 104.·29-s + 17.7·31-s + (−57.5 − 53.0i)35-s + 299. i·37-s − 467.·41-s + 34.2i·43-s − 199. i·47-s + ⋯
L(s)  = 1  + (0.677 − 0.735i)5-s − 0.377i·7-s − 1.59·11-s − 1.50i·13-s + 0.638i·17-s + 0.324·19-s − 0.171i·23-s + (−0.0823 − 0.996i)25-s − 0.667·29-s + 0.102·31-s + (−0.278 − 0.256i)35-s + 1.33i·37-s − 1.77·41-s + 0.121i·43-s − 0.619i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.735 - 0.677i$
Analytic conductor: \(74.3424\)
Root analytic conductor: \(8.62220\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :3/2),\ -0.735 - 0.677i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1313092247\)
\(L(\frac12)\) \(\approx\) \(0.1313092247\)
\(L(\frac{5}{2})\) 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 \( 1 + (-7.57 + 8.22i)T \)
7 \( 1 + 7iT \)
good11 \( 1 + 58.2T + 1.33e3T^{2} \)
13 \( 1 + 70.3iT - 2.19e3T^{2} \)
17 \( 1 - 44.7iT - 4.91e3T^{2} \)
19 \( 1 - 26.8T + 6.85e3T^{2} \)
23 \( 1 + 18.8iT - 1.21e4T^{2} \)
29 \( 1 + 104.T + 2.43e4T^{2} \)
31 \( 1 - 17.7T + 2.97e4T^{2} \)
37 \( 1 - 299. iT - 5.06e4T^{2} \)
41 \( 1 + 467.T + 6.89e4T^{2} \)
43 \( 1 - 34.2iT - 7.95e4T^{2} \)
47 \( 1 + 199. iT - 1.03e5T^{2} \)
53 \( 1 - 584. iT - 1.48e5T^{2} \)
59 \( 1 - 678.T + 2.05e5T^{2} \)
61 \( 1 + 628.T + 2.26e5T^{2} \)
67 \( 1 - 817. iT - 3.00e5T^{2} \)
71 \( 1 - 520.T + 3.57e5T^{2} \)
73 \( 1 + 417. iT - 3.89e5T^{2} \)
79 \( 1 - 375.T + 4.93e5T^{2} \)
83 \( 1 + 367. iT - 5.71e5T^{2} \)
89 \( 1 + 414.T + 7.04e5T^{2} \)
97 \( 1 - 762. iT - 9.12e5T^{2} \)
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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

−8.615637592879936188597647846798, −8.129695404598342214387860051718, −7.32618589064341882473815001661, −6.08659780308148549718065712005, −5.37260931009846798663669034037, −4.77549749031232578044846195554, −3.40870781205256989002022583434, −2.43019899286941032532091952201, −1.17290056939743858193682682596, −0.02964197501155196524420197363, 1.84994128538031236005343706884, 2.54796051323242177618993907422, 3.59621954586283896651441333910, 4.96574430858866185204388961497, 5.56168462811386510090538029672, 6.60357044786594700566271684091, 7.24764697334846110878952707802, 8.151118101473109660656619569537, 9.222934786751059123565510667862, 9.723972652903944041831546668380

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