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Properties

Label 2-1260-7.2-c1-0-7
Degree $2$
Conductor $1260$
Sign $0.895 + 0.444i$
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 + (1.32 − 2.29i)7-s + (1.82 + 3.15i)11-s + 2.64·13-s + (1.82 + 3.15i)17-s + (1.14 − 1.98i)19-s + (−1.82 + 3.15i)23-s + (−0.499 − 0.866i)25-s − 2.35·29-s + (−3.14 − 5.44i)31-s + (−1.32 − 2.29i)35-s + (2.32 − 4.02i)37-s + 10.9·41-s + 9.93·43-s + (3 − 5.19i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.499 − 0.866i)7-s + (0.549 + 0.951i)11-s + 0.733·13-s + (0.442 + 0.765i)17-s + (0.262 − 0.455i)19-s + (−0.380 + 0.658i)23-s + (−0.0999 − 0.173i)25-s − 0.437·29-s + (−0.564 − 0.978i)31-s + (−0.223 − 0.387i)35-s + (0.381 − 0.661i)37-s + 1.70·41-s + 1.51·43-s + (0.437 − 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 + 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.895 + 0.444i$
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: \((2,\ 1260,\ (\ :1/2),\ 0.895 + 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.968458277\)
\(L(\frac12)\) \(\approx\) \(1.968458277\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 + 2.29i)T \)
good11 \( 1 + (-1.82 - 3.15i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 + (-1.82 - 3.15i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.14 + 1.98i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.82 - 3.15i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.35T + 29T^{2} \)
31 \( 1 + (3.14 + 5.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.32 + 4.02i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 10.9T + 41T^{2} \)
43 \( 1 - 9.93T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.64 + 6.31i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.46 + 4.27i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.32 - 4.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.35T + 71T^{2} \)
73 \( 1 + (-6.61 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.14 + 7.18i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.93T + 83T^{2} \)
89 \( 1 + (-6.11 + 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−9.556286422419796621092715305505, −8.960347558840496830936748774436, −7.80563344608795027497130913434, −7.38941703503139042160467916727, −6.25347578913483579641513818144, −5.44019881541301818436715284472, −4.28497816259393512866648339923, −3.79662224238735448461526805623, −2.09667361890457538878958205230, −1.05728629360491220302824439036, 1.22800476100533845974334487086, 2.57417255265591881432418633395, 3.48873397837469220704955005331, 4.65858099046749551853078638757, 5.85417019853883451977584433147, 6.10183484186941932344513029080, 7.38122139972552554006121466878, 8.170679536900099714137701244292, 9.020385647097935025823120173543, 9.525469091439700638519961138507

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