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Properties

Label 2-3240-360.133-c0-0-3
Degree $2$
Conductor $3240$
Sign $0.313 + 0.949i$
Analytic cond. $1.61697$
Root an. cond. $1.27160$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.258 + 0.965i)5-s + (−0.133 − 0.5i)7-s + (−0.707 + 0.707i)8-s + (−0.499 − 0.866i)10-s + (−1.67 − 0.965i)11-s + (0.258 + 0.448i)14-s + (0.500 − 0.866i)16-s + (0.707 + 0.707i)20-s + (1.86 + 0.500i)22-s + (−0.866 + 0.499i)25-s + (−0.366 − 0.366i)28-s + (0.707 − 1.22i)29-s + (−0.866 − 1.5i)31-s + (−0.258 + 0.965i)32-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.258 + 0.965i)5-s + (−0.133 − 0.5i)7-s + (−0.707 + 0.707i)8-s + (−0.499 − 0.866i)10-s + (−1.67 − 0.965i)11-s + (0.258 + 0.448i)14-s + (0.500 − 0.866i)16-s + (0.707 + 0.707i)20-s + (1.86 + 0.500i)22-s + (−0.866 + 0.499i)25-s + (−0.366 − 0.366i)28-s + (0.707 − 1.22i)29-s + (−0.866 − 1.5i)31-s + (−0.258 + 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.313 + 0.949i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (3133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ 0.313 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5100033044\)
\(L(\frac12)\) \(\approx\) \(0.5100033044\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-0.258 - 0.965i)T \)
good7 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
59 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 1.86i)T + (-0.866 + 0.5i)T^{2} \)
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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

−8.564168983084622936070155593462, −7.80200043781023396035797735990, −7.47059491684325714755269589437, −6.49680032649228082551416298087, −5.91364381216310237597444407887, −5.18902003714662341840058400989, −3.73920599516529649697697457417, −2.78974269395531769440838926040, −2.13934129965676277686445515647, −0.41485167426543318391160694198, 1.33432078542802351284430707830, 2.30769179376819711998098245244, 3.07519926355356952610075763486, 4.43102844111491292429304481759, 5.26340237420189314866135298885, 5.90942277307252937554351830298, 7.11800959754255889729714828065, 7.54219336686683672713084089950, 8.568076904365053112775317654855, 8.810917917400518717085035600030

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