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Properties

Label 2-120-120.77-c1-0-12
Degree $2$
Conductor $120$
Sign $-0.793 + 0.608i$
Analytic cond. $0.958204$
Root an. cond. $0.978879$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 − i)2-s + (−1.22 + 1.22i)3-s + 2i·4-s + (−0.224 − 2.22i)5-s + 2.44·6-s + (−3.44 − 3.44i)7-s + (2 − 2i)8-s − 2.99i·9-s + (−2 + 2.44i)10-s − 1.55·11-s + (−2.44 − 2.44i)12-s + 6.89i·14-s + (2.99 + 2.44i)15-s − 4·16-s + (−2.99 + 2.99i)18-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + i·4-s + (−0.100 − 0.994i)5-s + 0.999·6-s + (−1.30 − 1.30i)7-s + (0.707 − 0.707i)8-s − 0.999i·9-s + (−0.632 + 0.774i)10-s − 0.467·11-s + (−0.707 − 0.707i)12-s + 1.84i·14-s + (0.774 + 0.632i)15-s − 16-s + (−0.707 + 0.707i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $-0.793 + 0.608i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ -0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113642 - 0.334915i\)
\(L(\frac12)\) \(\approx\) \(0.113642 - 0.334915i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 + i)T \)
3 \( 1 + (1.22 - 1.22i)T \)
5 \( 1 + (0.224 + 2.22i)T \)
good7 \( 1 + (3.44 + 3.44i)T + 7iT^{2} \)
11 \( 1 + 1.55T + 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 + 5.34iT - 29T^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 + (2.44 - 2.44i)T - 53iT^{2} \)
59 \( 1 + 15.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-11.8 + 11.8i)T - 73iT^{2} \)
79 \( 1 + 14.6iT - 79T^{2} \)
83 \( 1 + (-4 + 4i)T - 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-8.79 - 8.79i)T + 97iT^{2} \)
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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

−12.84094295954931632712174178585, −11.94380057154034893036654790062, −10.76510175032595696985505584993, −9.952007263840718969119806461024, −9.284810967207230374073333883891, −7.83810298226112589553678994665, −6.43854356893763885829647408399, −4.60347384478523713364864760544, −3.53363303668574277109226204230, −0.50524988570454925015104953906, 2.57727608679077880265242898249, 5.47280225280286794693353628525, 6.33779383600398662852933774617, 7.10133167012079581071248476374, 8.383875159524007905318653143328, 9.702696998792947270740110740771, 10.64359218870153862687237937803, 11.74789055255361908115407160818, 12.82990784968328611677901963518, 13.95398847255071077782654094180

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