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Properties

Label 2-60e2-180.83-c0-0-0
Degree $2$
Conductor $3600$
Sign $0.929 - 0.370i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
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)3-s + (−0.965 + 0.258i)7-s + (0.866 + 0.499i)9-s + 21-s + (0.448 − 1.67i)23-s + (−0.707 − 0.707i)27-s + (−0.866 + 1.5i)29-s + (1.5 − 0.866i)41-s + (0.517 + 1.93i)43-s + (0.448 + 1.67i)47-s + (0.5 − 0.866i)61-s + (−0.965 − 0.258i)63-s + (−0.258 + 0.965i)67-s + (−0.866 + 1.50i)69-s + (0.500 + 0.866i)81-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)3-s + (−0.965 + 0.258i)7-s + (0.866 + 0.499i)9-s + 21-s + (0.448 − 1.67i)23-s + (−0.707 − 0.707i)27-s + (−0.866 + 1.5i)29-s + (1.5 − 0.866i)41-s + (0.517 + 1.93i)43-s + (0.448 + 1.67i)47-s + (0.5 − 0.866i)61-s + (−0.965 − 0.258i)63-s + (−0.258 + 0.965i)67-s + (−0.866 + 1.50i)69-s + (0.500 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.370i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.929 - 0.370i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.929 - 0.370i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7424513731\)
\(L(\frac12)\) \(\approx\) \(0.7424513731\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 \)
good7 \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.448 + 1.67i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.517 - 1.93i)T + (-0.866 + 0.5i)T^{2} \)
47 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.67 + 0.448i)T + (0.866 - 0.5i)T^{2} \)
89 \( 1 - 1.73T + T^{2} \)
97 \( 1 + (-0.866 + 0.5i)T^{2} \)
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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

−8.967436855342813232473744949365, −7.83668684619926911052837152039, −7.18809851337611607863616732125, −6.35281362802412855457198206054, −6.02997559914578599469721700698, −5.03477627746774272799992504523, −4.34192343451179237013942811989, −3.24811508943558638549995823779, −2.27961367111434816480953193505, −0.913802526597255475115959050938, 0.66219421381688249753656634920, 2.10002569668560305218844109179, 3.47774591438695923146296624137, 3.96307509158137507949747923073, 5.02517063460891729457339787551, 5.77143523588878455025184997159, 6.30297614095968244156255688837, 7.20760086921808955417549723978, 7.64077999288035285999157642068, 8.943245569434132189248636260479

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