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Properties

Label 2-60-60.23-c2-0-16
Degree $2$
Conductor $60$
Sign $-0.481 + 0.876i$
Analytic cond. $1.63488$
Root an. cond. $1.27862$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.07 − 1.68i)2-s + (−2.99 − 0.130i)3-s + (−1.67 − 3.63i)4-s + (1.65 − 4.71i)5-s + (−3.45 + 4.90i)6-s + (−1.91 − 1.91i)7-s + (−7.92 − 1.11i)8-s + (8.96 + 0.782i)9-s + (−6.16 − 7.87i)10-s + 6.87·11-s + (4.53 + 11.1i)12-s + (12.2 + 12.2i)13-s + (−5.29 + 1.15i)14-s + (−5.56 + 13.9i)15-s + (−10.4 + 12.1i)16-s + (−9.47 − 9.47i)17-s + ⋯
L(s)  = 1  + (0.539 − 0.841i)2-s + (−0.999 − 0.0434i)3-s + (−0.417 − 0.908i)4-s + (0.330 − 0.943i)5-s + (−0.575 + 0.817i)6-s + (−0.273 − 0.273i)7-s + (−0.990 − 0.138i)8-s + (0.996 + 0.0869i)9-s + (−0.616 − 0.787i)10-s + 0.624·11-s + (0.377 + 0.925i)12-s + (0.944 + 0.944i)13-s + (−0.378 + 0.0826i)14-s + (−0.370 + 0.928i)15-s + (−0.651 + 0.758i)16-s + (−0.557 − 0.557i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.481 + 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $-0.481 + 0.876i$
Analytic conductor: \(1.63488\)
Root analytic conductor: \(1.27862\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 60,\ (\ :1),\ -0.481 + 0.876i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.581404 - 0.983387i\)
\(L(\frac12)\) \(\approx\) \(0.581404 - 0.983387i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 + 1.68i)T \)
3 \( 1 + (2.99 + 0.130i)T \)
5 \( 1 + (-1.65 + 4.71i)T \)
good7 \( 1 + (1.91 + 1.91i)T + 49iT^{2} \)
11 \( 1 - 6.87T + 121T^{2} \)
13 \( 1 + (-12.2 - 12.2i)T + 169iT^{2} \)
17 \( 1 + (9.47 + 9.47i)T + 289iT^{2} \)
19 \( 1 - 33.2T + 361T^{2} \)
23 \( 1 + (7.20 + 7.20i)T + 529iT^{2} \)
29 \( 1 + 2.29T + 841T^{2} \)
31 \( 1 - 12.1iT - 961T^{2} \)
37 \( 1 + (20.7 - 20.7i)T - 1.36e3iT^{2} \)
41 \( 1 - 50.9iT - 1.68e3T^{2} \)
43 \( 1 + (-15.1 + 15.1i)T - 1.84e3iT^{2} \)
47 \( 1 + (-26.7 + 26.7i)T - 2.20e3iT^{2} \)
53 \( 1 + (15.5 - 15.5i)T - 2.80e3iT^{2} \)
59 \( 1 - 63.0iT - 3.48e3T^{2} \)
61 \( 1 - 28.4T + 3.72e3T^{2} \)
67 \( 1 + (-32.4 - 32.4i)T + 4.48e3iT^{2} \)
71 \( 1 + 88.8T + 5.04e3T^{2} \)
73 \( 1 + (-71.1 - 71.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 75.1T + 6.24e3T^{2} \)
83 \( 1 + (-58.6 - 58.6i)T + 6.88e3iT^{2} \)
89 \( 1 - 41.1T + 7.92e3T^{2} \)
97 \( 1 + (-30.3 + 30.3i)T - 9.40e3iT^{2} \)
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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

−13.94308625426378472682124141619, −13.26270860808168332354400195657, −12.03322097532764226432532049969, −11.43502322307076476756752435720, −10.01699885221616429610407088592, −9.048881917483067411203453017823, −6.62749228902110573969134523562, −5.35967214104520515901017419599, −4.12210677881067079528393131488, −1.22804596149007361207738155685, 3.61514067485228076476557045323, 5.55957523324856945726330039970, 6.35538786272521071922838590780, 7.55026039884938407441635049979, 9.402392440581952797733908920410, 10.83637871643918081917632006938, 11.93924183507152329984299835432, 13.12272824598011597816042028248, 14.14507565092444289899106639225, 15.46072214486667373097255222668

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