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Properties

Label 2-60-12.11-c3-0-7
Degree $2$
Conductor $60$
Sign $0.864 - 0.503i$
Analytic cond. $3.54011$
Root an. cond. $1.88151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.72 − 0.763i)2-s + (4.56 + 2.47i)3-s + (6.83 + 4.15i)4-s − 5i·5-s + (−10.5 − 10.2i)6-s + 20.9i·7-s + (−15.4 − 16.5i)8-s + (14.7 + 22.6i)9-s + (−3.81 + 13.6i)10-s + 30.8·11-s + (20.9 + 35.9i)12-s + 56.7·13-s + (16.0 − 57.1i)14-s + (12.3 − 22.8i)15-s + (29.4 + 56.8i)16-s + 88.9i·17-s + ⋯
L(s)  = 1  + (−0.962 − 0.269i)2-s + (0.878 + 0.476i)3-s + (0.854 + 0.519i)4-s − 0.447i·5-s + (−0.717 − 0.696i)6-s + 1.13i·7-s + (−0.682 − 0.730i)8-s + (0.544 + 0.838i)9-s + (−0.120 + 0.430i)10-s + 0.844·11-s + (0.503 + 0.864i)12-s + 1.21·13-s + (0.305 − 1.09i)14-s + (0.213 − 0.393i)15-s + (0.460 + 0.887i)16-s + 1.26i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.864 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.864 - 0.503i$
Analytic conductor: \(3.54011\)
Root analytic conductor: \(1.88151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :3/2),\ 0.864 - 0.503i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.23561 + 0.333507i\)
\(L(\frac12)\) \(\approx\) \(1.23561 + 0.333507i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.72 + 0.763i)T \)
3 \( 1 + (-4.56 - 2.47i)T \)
5 \( 1 + 5iT \)
good7 \( 1 - 20.9iT - 343T^{2} \)
11 \( 1 - 30.8T + 1.33e3T^{2} \)
13 \( 1 - 56.7T + 2.19e3T^{2} \)
17 \( 1 - 88.9iT - 4.91e3T^{2} \)
19 \( 1 + 88.2iT - 6.85e3T^{2} \)
23 \( 1 + 138.T + 1.21e4T^{2} \)
29 \( 1 + 161. iT - 2.43e4T^{2} \)
31 \( 1 + 197. iT - 2.97e4T^{2} \)
37 \( 1 + 179.T + 5.06e4T^{2} \)
41 \( 1 + 67.7iT - 6.89e4T^{2} \)
43 \( 1 + 41.7iT - 7.95e4T^{2} \)
47 \( 1 + 214.T + 1.03e5T^{2} \)
53 \( 1 + 263. iT - 1.48e5T^{2} \)
59 \( 1 - 103.T + 2.05e5T^{2} \)
61 \( 1 + 698.T + 2.26e5T^{2} \)
67 \( 1 + 129. iT - 3.00e5T^{2} \)
71 \( 1 - 301.T + 3.57e5T^{2} \)
73 \( 1 - 1.11e3T + 3.89e5T^{2} \)
79 \( 1 + 712. iT - 4.93e5T^{2} \)
83 \( 1 - 336.T + 5.71e5T^{2} \)
89 \( 1 - 970. iT - 7.04e5T^{2} \)
97 \( 1 + 1.60e3T + 9.12e5T^{2} \)
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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

−15.12343559628596684186591208036, −13.51094767977376026980091524446, −12.25719258282709201864012060599, −11.08302027189286948443135921940, −9.689632255017984469255539638608, −8.812138728710080635552175547722, −8.128827192870194067844068622049, −6.17204357743657437175770407672, −3.81324197092356422562061313504, −1.99389900144814828239771607427, 1.36470811850134153567936354558, 3.52125700363213873662104764241, 6.44032548888170059244686363025, 7.34754851997697096219224075104, 8.469158630128365920297832964347, 9.670771171028709636657776759046, 10.75884709490582951263900109995, 12.09443215683876404336213544462, 13.91109181600999644112648549366, 14.27785813967977103736642285036

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