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Properties

Label 2-1260-7.4-c1-0-12
Degree $2$
Conductor $1260$
Sign $-0.991 + 0.126i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)5-s + (0.5 + 2.59i)7-s + (−1 + 1.73i)11-s − 6·13-s + (1 − 1.73i)17-s + (−4.5 − 7.79i)23-s + (−0.499 + 0.866i)25-s − 3·29-s + (−1 + 1.73i)31-s + (2 − 1.73i)35-s + (−4 − 6.92i)37-s − 5·41-s + 43-s + (4 + 6.92i)47-s + (−6.5 + 2.59i)49-s + ⋯
L(s)  = 1  + (−0.223 − 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.301 + 0.522i)11-s − 1.66·13-s + (0.242 − 0.420i)17-s + (−0.938 − 1.62i)23-s + (−0.0999 + 0.173i)25-s − 0.557·29-s + (−0.179 + 0.311i)31-s + (0.338 − 0.292i)35-s + (−0.657 − 1.13i)37-s − 0.780·41-s + 0.152·43-s + (0.583 + 1.01i)47-s + (−0.928 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.991 + 0.126i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 2.59i)T \)
good11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4 + 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.5 + 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 + 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - T + 83T^{2} \)
89 \( 1 + (-6.5 - 11.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.309868891136662265744523179206, −8.511392212267655659554470876859, −7.69402221458071211565812372635, −6.95589143197288856539100214003, −5.77308167799529509719522802573, −5.02780948873472597040228352468, −4.29768152399406475569215937741, −2.79670213459983831028136736420, −2.00867417997297657111575291231, 0, 1.75206441701054846215614564285, 3.10052016296613180675074930278, 3.96442610923289746335142871840, 4.98181939404984270510884788997, 5.87907712310243462831817296094, 7.05501691172677050728245760253, 7.53815653818129945143118025082, 8.243255835849661930905107090134, 9.461088331473120875808656168144

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