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Properties

Label 2-1260-28.27-c1-0-13
Degree $2$
Conductor $1260$
Sign $-0.996 + 0.0891i$
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  + (0.449 + 1.34i)2-s + (−1.59 + 1.20i)4-s + i·5-s + (1.40 − 2.24i)7-s + (−2.33 − 1.59i)8-s + (−1.34 + 0.449i)10-s + 3.99i·11-s + 2.50i·13-s + (3.63 + 0.868i)14-s + (1.09 − 3.84i)16-s − 1.07i·17-s − 7.78·19-s + (−1.20 − 1.59i)20-s + (−5.35 + 1.79i)22-s + 8.10i·23-s + ⋯
L(s)  = 1  + (0.317 + 0.948i)2-s + (−0.797 + 0.602i)4-s + 0.447i·5-s + (0.529 − 0.848i)7-s + (−0.825 − 0.564i)8-s + (−0.424 + 0.142i)10-s + 1.20i·11-s + 0.694i·13-s + (0.972 + 0.232i)14-s + (0.273 − 0.961i)16-s − 0.261i·17-s − 1.78·19-s + (−0.269 − 0.356i)20-s + (−1.14 + 0.382i)22-s + 1.69i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.996 + 0.0891i$
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: \((2,\ 1260,\ (\ :1/2),\ -0.996 + 0.0891i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.197785243\)
\(L(\frac12)\) \(\approx\) \(1.197785243\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.449 - 1.34i)T \)
3 \( 1 \)
5 \( 1 - iT \)
7 \( 1 + (-1.40 + 2.24i)T \)
good11 \( 1 - 3.99iT - 11T^{2} \)
13 \( 1 - 2.50iT - 13T^{2} \)
17 \( 1 + 1.07iT - 17T^{2} \)
19 \( 1 + 7.78T + 19T^{2} \)
23 \( 1 - 8.10iT - 23T^{2} \)
29 \( 1 - 5.99T + 29T^{2} \)
31 \( 1 + 6.32T + 31T^{2} \)
37 \( 1 - 3.05T + 37T^{2} \)
41 \( 1 - 9.32iT - 41T^{2} \)
43 \( 1 - 8.12iT - 43T^{2} \)
47 \( 1 + 9.01T + 47T^{2} \)
53 \( 1 + 2.86T + 53T^{2} \)
59 \( 1 + 9.68T + 59T^{2} \)
61 \( 1 + 3.21iT - 61T^{2} \)
67 \( 1 + 7.79iT - 67T^{2} \)
71 \( 1 + 5.84iT - 71T^{2} \)
73 \( 1 - 5.73iT - 73T^{2} \)
79 \( 1 + 2.81iT - 79T^{2} \)
83 \( 1 - 12.5T + 83T^{2} \)
89 \( 1 - 3.51iT - 89T^{2} \)
97 \( 1 + 12.0iT - 97T^{2} \)
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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

−9.864107331931179560421440773635, −9.311032792524670615325880151307, −8.133402838682563229721333301126, −7.58155349442346511702441707388, −6.79670996060709908655032828612, −6.21257374395906148583334363967, −4.80564066657634632865204904799, −4.42881277655298202533990692229, −3.35018751298859552556508519694, −1.78820032501911489030959540573, 0.44373388399373969080077418917, 1.94406656106638848542466004173, 2.85631095581082696836975695449, 4.01130339713533794307445421459, 4.89812323950380586334458672916, 5.72629709761943764981446598584, 6.41122908769819545972202426040, 8.191620376493871887864744421543, 8.560543959593004505267557935894, 9.138761822677444309744995326784

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