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Properties

Label 2-1260-45.34-c1-0-6
Degree $2$
Conductor $1260$
Sign $-0.803 - 0.595i$
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  + (0.866 + 1.5i)3-s + (−0.133 − 2.23i)5-s + (−0.866 − 0.5i)7-s + (−1.5 + 2.59i)9-s + (−2 + 3.46i)11-s + (−2.59 + 1.5i)13-s + (3.23 − 2.13i)15-s + i·17-s − 1.73i·21-s + (1.73 − i)23-s + (−4.96 + 0.598i)25-s − 5.19·27-s + (−4.5 + 7.79i)29-s + (1 + 1.73i)31-s − 6.92·33-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s + (−0.0599 − 0.998i)5-s + (−0.327 − 0.188i)7-s + (−0.5 + 0.866i)9-s + (−0.603 + 1.04i)11-s + (−0.720 + 0.416i)13-s + (0.834 − 0.550i)15-s + 0.242i·17-s − 0.377i·21-s + (0.361 − 0.208i)23-s + (−0.992 + 0.119i)25-s − 1.00·27-s + (−0.835 + 1.44i)29-s + (0.179 + 0.311i)31-s − 1.20·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.803 - 0.595i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.803 - 0.595i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9331285820\)
\(L(\frac12)\) \(\approx\) \(0.9331285820\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T \)
5 \( 1 + (0.133 + 2.23i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
good11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.59 - 1.5i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-1.73 + i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.5 - 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (-4 - 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.46 + 2i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.92 + 4i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.92 - 4i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 - iT - 73T^{2} \)
79 \( 1 + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.33 + 2.5i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + (8.66 + 5i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.796313593029873349018238118791, −9.347859833471646882739648034746, −8.515220318965264926188194603035, −7.73798440854763581377649924047, −6.86897342765211006097996292783, −5.44698358171523509653490043032, −4.82464740376244242357242108563, −4.13759729138203064746141066635, −2.97679726476481985213690496002, −1.77658687271086210975139644633, 0.34132588534484034179728540804, 2.22653841121348603991933940973, 2.92829581487723446817933879150, 3.76084035316019385047430603415, 5.43078662652655704820051166546, 6.15153611542406303321877028999, 6.98723521080913504598625588893, 7.72863839283245781324370464610, 8.309221685403433584603119277220, 9.402151994899502498072247510547

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