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Properties

Label 2-60e2-300.227-c0-0-0
Degree $2$
Conductor $3600$
Sign $-0.675 - 0.737i$
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  + (−0.453 + 0.891i)5-s + (−1.76 + 0.278i)13-s + (1.44 + 0.734i)17-s + (−0.587 − 0.809i)25-s + (−0.610 + 1.87i)29-s + (−0.309 − 1.95i)37-s + (−1.04 + 1.44i)41-s + i·49-s + (−0.550 + 0.280i)53-s + (−1.53 + 1.11i)61-s + (0.550 − 1.69i)65-s + (−0.142 + 0.896i)73-s + (−1.30 + 0.951i)85-s + (−0.253 + 0.183i)89-s + (0.278 − 0.142i)97-s + ⋯
L(s)  = 1  + (−0.453 + 0.891i)5-s + (−1.76 + 0.278i)13-s + (1.44 + 0.734i)17-s + (−0.587 − 0.809i)25-s + (−0.610 + 1.87i)29-s + (−0.309 − 1.95i)37-s + (−1.04 + 1.44i)41-s + i·49-s + (−0.550 + 0.280i)53-s + (−1.53 + 1.11i)61-s + (0.550 − 1.69i)65-s + (−0.142 + 0.896i)73-s + (−1.30 + 0.951i)85-s + (−0.253 + 0.183i)89-s + (0.278 − 0.142i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.675 - 0.737i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ -0.675 - 0.737i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7027005338\)
\(L(\frac12)\) \(\approx\) \(0.7027005338\)
\(L(1)\) 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 \( 1 \)
5 \( 1 + (0.453 - 0.891i)T \)
good7 \( 1 - iT^{2} \)
11 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (1.76 - 0.278i)T + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (-1.44 - 0.734i)T + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.951 - 0.309i)T^{2} \)
29 \( 1 + (0.610 - 1.87i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 + 1.95i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (1.04 - 1.44i)T + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (0.550 - 0.280i)T + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.587 + 0.809i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (-0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (0.253 - 0.183i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
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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

−9.065872590420156110316806067095, −8.055120661587771696154837865583, −7.41368036722546551030132712995, −7.04069231982499765340247564723, −6.03100406563423765197737759734, −5.26384353912823794404353681942, −4.37568612840633541214864099643, −3.42633169575533134871332685543, −2.77738322998514131513629487752, −1.64030205045955792117926725812, 0.39373198153570822192600322678, 1.79984496134697849913559271708, 2.92913816115938371714763314059, 3.81013369426039084546643391126, 4.94982358032129091894977546307, 5.09172152690693227409554193800, 6.15161938155399734700563017889, 7.25444928210178426058412990594, 7.72676189778306841944401503158, 8.313036634374430151813807499282

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