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Properties

Label 2-360-40.13-c0-0-0
Degree $2$
Conductor $360$
Sign $0.525 - 0.850i$
Analytic cond. $0.179663$
Root an. cond. $0.423867$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−1 + i)7-s + (0.707 − 0.707i)8-s + 1.00·10-s + 1.41i·11-s + 1.41·14-s − 1.00·16-s + (−0.707 − 0.707i)20-s + (1.00 − 1.00i)22-s − 1.00i·25-s + (−1.00 − 1.00i)28-s + 1.41·29-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)5-s + (−1 + i)7-s + (0.707 − 0.707i)8-s + 1.00·10-s + 1.41i·11-s + 1.41·14-s − 1.00·16-s + (−0.707 − 0.707i)20-s + (1.00 − 1.00i)22-s − 1.00i·25-s + (−1.00 − 1.00i)28-s + 1.41·29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(360\)    =    \(2^{3} \cdot 3^{2} \cdot 5\)
Sign: $0.525 - 0.850i$
Analytic conductor: \(0.179663\)
Root analytic conductor: \(0.423867\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{360} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 360,\ (\ :0),\ 0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4244633853\)
\(L(\frac12)\) \(\approx\) \(0.4244633853\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1 - i)T - iT^{2} \)
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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

−12.00783858044549527177114152635, −10.78342271627474309140749630543, −9.977242364606914542759665771846, −9.255860202726803381749867046410, −8.208546836024751702893021789920, −7.21096218854956386094139321847, −6.39410761104208663062318913002, −4.56666170557139465868655513850, −3.29027235777017944307659118010, −2.33639757280357245296855103978, 0.78444490821900361124479616680, 3.39730127438697684948271408656, 4.65881908694951234176676278767, 5.95717546530971379593507919447, 6.85260742301055121260695771619, 7.86772973203799178124517426089, 8.614672253269514681248548188284, 9.509274170574110250484335096006, 10.49127867037549412346363878360, 11.24420042130513072719694029754

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