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Properties

Label 2-1260-35.3-c1-0-1
Degree $2$
Conductor $1260$
Sign $-0.540 - 0.841i$
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  + (−1.33 − 1.79i)5-s + (−0.921 − 2.47i)7-s + (−2.62 + 4.54i)11-s + (0.0789 − 0.0789i)13-s + (0.566 − 0.151i)17-s + (−1.74 − 3.01i)19-s + (−0.849 + 3.17i)23-s + (−1.42 + 4.79i)25-s + 1.60i·29-s + (−6.78 − 3.91i)31-s + (−3.20 + 4.96i)35-s + (5.08 + 1.36i)37-s + 9.92i·41-s + (6.41 + 6.41i)43-s + (−0.795 + 2.96i)47-s + ⋯
L(s)  = 1  + (−0.598 − 0.801i)5-s + (−0.348 − 0.937i)7-s + (−0.791 + 1.37i)11-s + (0.0219 − 0.0219i)13-s + (0.137 − 0.0368i)17-s + (−0.399 − 0.691i)19-s + (−0.177 + 0.661i)23-s + (−0.284 + 0.958i)25-s + 0.297i·29-s + (−1.21 − 0.703i)31-s + (−0.542 + 0.840i)35-s + (0.836 + 0.224i)37-s + 1.55i·41-s + (0.979 + 0.979i)43-s + (−0.116 + 0.433i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.540 - 0.841i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.540 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2867576338\)
\(L(\frac12)\) \(\approx\) \(0.2867576338\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.33 + 1.79i)T \)
7 \( 1 + (0.921 + 2.47i)T \)
good11 \( 1 + (2.62 - 4.54i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.0789 + 0.0789i)T - 13iT^{2} \)
17 \( 1 + (-0.566 + 0.151i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.74 + 3.01i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.849 - 3.17i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.60iT - 29T^{2} \)
31 \( 1 + (6.78 + 3.91i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.08 - 1.36i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 9.92iT - 41T^{2} \)
43 \( 1 + (-6.41 - 6.41i)T + 43iT^{2} \)
47 \( 1 + (0.795 - 2.96i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.98 - 1.33i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.80 - 3.12i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.98 + 1.14i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.681 - 2.54i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.46T + 71T^{2} \)
73 \( 1 + (-0.453 - 1.69i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.23 - 0.710i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.68 - 8.68i)T - 83iT^{2} \)
89 \( 1 + (6.39 + 11.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.289 - 0.289i)T + 97iT^{2} \)
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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

−9.724328155417660106847635248331, −9.409625027360072203981613048667, −8.126246927748536572088907509657, −7.57913736035496438259786022231, −6.91421112800232270822179714744, −5.67719500051742185961562627748, −4.61649410646895587615868561716, −4.17891433585797042958417455848, −2.88410670012411399014298880341, −1.40300751997427725388029560920, 0.12092992611779901092538770073, 2.27916062236268924452320214290, 3.15148214230781449067239308357, 3.97420934291796319895197403067, 5.43550747480892492510398411980, 5.99326784277622352969827573520, 6.91209223381591335450637529856, 7.914820072191319093023910247891, 8.497989576111543506058449005236, 9.314424632010597554556834232231

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