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Properties

Label 2-60-12.11-c5-0-3
Degree $2$
Conductor $60$
Sign $-0.0799 - 0.996i$
Analytic cond. $9.62302$
Root an. cond. $3.10210$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−5.57 − 0.966i)2-s + (−14.2 + 6.40i)3-s + (30.1 + 10.7i)4-s − 25i·5-s + (85.4 − 21.9i)6-s − 68.6i·7-s + (−157. − 89.1i)8-s + (160. − 182. i)9-s + (−24.1 + 139. i)10-s + 118.·11-s + (−497. + 39.8i)12-s − 360.·13-s + (−66.3 + 382. i)14-s + (160. + 355. i)15-s + (791. + 649. i)16-s + 1.87e3i·17-s + ⋯
L(s)  = 1  + (−0.985 − 0.170i)2-s + (−0.911 + 0.410i)3-s + (0.941 + 0.336i)4-s − 0.447i·5-s + (0.968 − 0.249i)6-s − 0.529i·7-s + (−0.870 − 0.492i)8-s + (0.662 − 0.749i)9-s + (−0.0763 + 0.440i)10-s + 0.296·11-s + (−0.996 + 0.0799i)12-s − 0.590·13-s + (−0.0904 + 0.521i)14-s + (0.183 + 0.407i)15-s + (0.773 + 0.633i)16-s + 1.57i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0799 - 0.996i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0799 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.0799 - 0.996i$
Analytic conductor: \(9.62302\)
Root analytic conductor: \(3.10210\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :5/2),\ -0.0799 - 0.996i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.321254 + 0.348043i\)
\(L(\frac12)\) \(\approx\) \(0.321254 + 0.348043i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.57 + 0.966i)T \)
3 \( 1 + (14.2 - 6.40i)T \)
5 \( 1 + 25iT \)
good7 \( 1 + 68.6iT - 1.68e4T^{2} \)
11 \( 1 - 118.T + 1.61e5T^{2} \)
13 \( 1 + 360.T + 3.71e5T^{2} \)
17 \( 1 - 1.87e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.31e3iT - 2.47e6T^{2} \)
23 \( 1 + 626.T + 6.43e6T^{2} \)
29 \( 1 - 5.24e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.58e3iT - 2.86e7T^{2} \)
37 \( 1 + 9.44e3T + 6.93e7T^{2} \)
41 \( 1 - 1.20e4iT - 1.15e8T^{2} \)
43 \( 1 + 3.92e3iT - 1.47e8T^{2} \)
47 \( 1 - 1.47e4T + 2.29e8T^{2} \)
53 \( 1 - 2.25e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.14e4T + 7.14e8T^{2} \)
61 \( 1 + 3.16e4T + 8.44e8T^{2} \)
67 \( 1 - 6.90e4iT - 1.35e9T^{2} \)
71 \( 1 + 7.63e4T + 1.80e9T^{2} \)
73 \( 1 - 6.70e4T + 2.07e9T^{2} \)
79 \( 1 - 7.96e4iT - 3.07e9T^{2} \)
83 \( 1 + 6.30e4T + 3.93e9T^{2} \)
89 \( 1 + 1.27e5iT - 5.58e9T^{2} \)
97 \( 1 - 5.77e4T + 8.58e9T^{2} \)
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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

−14.72510667331179400413912682044, −12.74439691231944224850127653133, −11.94218791088175194585751081481, −10.68004035578816413661312049312, −9.977326299760232131827064830422, −8.638339603621720997115700521325, −7.17317271865933777638992002528, −5.83466702952529604783973510358, −3.95580654365451357320607953935, −1.31990489620805140238003996964, 0.37315368949679181650963974507, 2.34521183732070387153668641074, 5.29423364228499100623630974445, 6.64560642241326133463365761613, 7.52951340176840259341426866012, 9.163291126476363392703640622659, 10.32688767400383188320123813523, 11.50022519366957490032229944735, 12.13381499503215349714863098009, 13.84654803307125611949487509915

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