Properties

Label 2-120-120.53-c1-0-12
Degree $2$
Conductor $120$
Sign $0.395 + 0.918i$
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  + (−0.864 − 1.11i)2-s + (1.72 + 0.170i)3-s + (−0.506 + 1.93i)4-s + (−1.36 − 1.77i)5-s + (−1.29 − 2.07i)6-s + (2.06 − 2.06i)7-s + (2.60 − 1.10i)8-s + (2.94 + 0.586i)9-s + (−0.801 + 3.05i)10-s − 0.510·11-s + (−1.20 + 3.24i)12-s + (−0.750 + 0.750i)13-s + (−4.10 − 0.528i)14-s + (−2.05 − 3.28i)15-s + (−3.48 − 1.95i)16-s + (3.14 + 3.14i)17-s + ⋯
L(s)  = 1  + (−0.611 − 0.791i)2-s + (0.995 + 0.0982i)3-s + (−0.253 + 0.967i)4-s + (−0.610 − 0.791i)5-s + (−0.530 − 0.847i)6-s + (0.782 − 0.782i)7-s + (0.920 − 0.390i)8-s + (0.980 + 0.195i)9-s + (−0.253 + 0.967i)10-s − 0.153·11-s + (−0.346 + 0.937i)12-s + (−0.208 + 0.208i)13-s + (−1.09 − 0.141i)14-s + (−0.530 − 0.847i)15-s + (−0.871 − 0.489i)16-s + (0.763 + 0.763i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.395 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(120\)    =    \(2^{3} \cdot 3 \cdot 5\)
Sign: $0.395 + 0.918i$
Analytic conductor: \(0.958204\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{120} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 120,\ (\ :1/2),\ 0.395 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.845574 - 0.556445i\)
\(L(\frac12)\) \(\approx\) \(0.845574 - 0.556445i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.864 + 1.11i)T \)
3 \( 1 + (-1.72 - 0.170i)T \)
5 \( 1 + (1.36 + 1.77i)T \)
good7 \( 1 + (-2.06 + 2.06i)T - 7iT^{2} \)
11 \( 1 + 0.510T + 11T^{2} \)
13 \( 1 + (0.750 - 0.750i)T - 13iT^{2} \)
17 \( 1 + (-3.14 - 3.14i)T + 17iT^{2} \)
19 \( 1 + 6.01T + 19T^{2} \)
23 \( 1 + (2.54 - 2.54i)T - 23iT^{2} \)
29 \( 1 - 5.10iT - 29T^{2} \)
31 \( 1 + 4.56T + 31T^{2} \)
37 \( 1 + (-6.76 - 6.76i)T + 37iT^{2} \)
41 \( 1 + 4.24iT - 41T^{2} \)
43 \( 1 + (-5.95 + 5.95i)T - 43iT^{2} \)
47 \( 1 + (3.33 + 3.33i)T + 47iT^{2} \)
53 \( 1 + (5.75 + 5.75i)T + 53iT^{2} \)
59 \( 1 - 1.16iT - 59T^{2} \)
61 \( 1 - 4.92iT - 61T^{2} \)
67 \( 1 + (7.98 + 7.98i)T + 67iT^{2} \)
71 \( 1 + 5.09iT - 71T^{2} \)
73 \( 1 + (3.20 + 3.20i)T + 73iT^{2} \)
79 \( 1 - 7.31iT - 79T^{2} \)
83 \( 1 + (-4.77 - 4.77i)T + 83iT^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 + (-10.8 + 10.8i)T - 97iT^{2} \)
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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

−13.07575356014361216564450739078, −12.33843698538608811232358600111, −11.05721433120227687575969082345, −10.13062717116363787105953361738, −8.924053379382377827772981522659, −8.129620931866976062342404254090, −7.40899606030206882979898434142, −4.58393491567434553021721038160, −3.67350521242189916865851404123, −1.65821972945591665998912860896, 2.41525802165398809173460916080, 4.43505833331988756985439214053, 6.15437524606949484509816753875, 7.55436491646515469775094471586, 8.077077391820210010301810055668, 9.163279770214773512401904210740, 10.28906027937259771274038926657, 11.41353795739144578922975735956, 12.81927260987029629447980513978, 14.36702189235722449192447158452

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