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Properties

Label 2-1260-35.12-c1-0-11
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: $\chi_{1260} (397, \cdot )$
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

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

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