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Properties

Label 2-60e2-100.31-c0-0-1
Degree $2$
Conductor $3600$
Sign $0.637 + 0.770i$
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  + (0.587 − 0.809i)5-s + (0.5 − 1.53i)13-s + (1.53 + 1.11i)17-s + (−0.309 − 0.951i)25-s + (−1.53 + 1.11i)29-s + (0.190 − 0.587i)37-s + (0.363 − 1.11i)41-s + 49-s + (−0.951 + 0.690i)53-s + (−0.5 − 1.53i)61-s + (−0.951 − 1.30i)65-s + (0.5 + 1.53i)73-s + (1.80 − 0.587i)85-s + (−0.587 − 1.80i)89-s + (0.5 − 0.363i)97-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)5-s + (0.5 − 1.53i)13-s + (1.53 + 1.11i)17-s + (−0.309 − 0.951i)25-s + (−1.53 + 1.11i)29-s + (0.190 − 0.587i)37-s + (0.363 − 1.11i)41-s + 49-s + (−0.951 + 0.690i)53-s + (−0.5 − 1.53i)61-s + (−0.951 − 1.30i)65-s + (0.5 + 1.53i)73-s + (1.80 − 0.587i)85-s + (−0.587 − 1.80i)89-s + (0.5 − 0.363i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.637 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.637 + 0.770i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.637 + 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.491538894\)
\(L(\frac12)\) \(\approx\) \(1.491538894\)
\(L(1)\) 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 \( 1 + (-0.587 + 0.809i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
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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

−8.634645538723734877188175634893, −7.928415303651406930754436133120, −7.36004432837544540425262262356, −6.02599320117936753597049595904, −5.69878666478171140842910200569, −5.06112316946891337130104262141, −3.87668915477515962908998267055, −3.21793009496791806389229853546, −1.92708906652448441659525863001, −0.977557961465797295495772041483, 1.41737282729308541845354976111, 2.39859839011044621208651091764, 3.30106422423217029575374342755, 4.13633216876818177349587965202, 5.16233490095659198185152732386, 5.96568153634088584923879080575, 6.55436437929771469128905478995, 7.37102695531173457431326062567, 7.893111300711987573229854036254, 9.108364717096661180817865307297

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