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Properties

Label 2-240-80.29-c1-0-20
Degree $2$
Conductor $240$
Sign $0.877 + 0.479i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
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  + (1.40 + 0.112i)2-s + (0.707 − 0.707i)3-s + (1.97 + 0.316i)4-s + (−0.466 − 2.18i)5-s + (1.07 − 0.917i)6-s − 1.00·7-s + (2.74 + 0.667i)8-s − 1.00i·9-s + (−0.413 − 3.13i)10-s + (−1.89 + 1.89i)11-s + (1.61 − 1.17i)12-s + (−2.65 + 2.65i)13-s + (−1.40 − 0.112i)14-s + (−1.87 − 1.21i)15-s + (3.80 + 1.24i)16-s − 1.73i·17-s + ⋯
L(s)  = 1  + (0.996 + 0.0792i)2-s + (0.408 − 0.408i)3-s + (0.987 + 0.158i)4-s + (−0.208 − 0.977i)5-s + (0.439 − 0.374i)6-s − 0.378·7-s + (0.971 + 0.235i)8-s − 0.333i·9-s + (−0.130 − 0.991i)10-s + (−0.571 + 0.571i)11-s + (0.467 − 0.338i)12-s + (−0.737 + 0.737i)13-s + (−0.376 − 0.0299i)14-s + (−0.484 − 0.314i)15-s + (0.950 + 0.312i)16-s − 0.421i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.877 + 0.479i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.877 + 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21387 - 0.565020i\)
\(L(\frac12)\) \(\approx\) \(2.21387 - 0.565020i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 - 0.112i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.466 + 2.18i)T \)
good7 \( 1 + 1.00T + 7T^{2} \)
11 \( 1 + (1.89 - 1.89i)T - 11iT^{2} \)
13 \( 1 + (2.65 - 2.65i)T - 13iT^{2} \)
17 \( 1 + 1.73iT - 17T^{2} \)
19 \( 1 + (-5.33 - 5.33i)T + 19iT^{2} \)
23 \( 1 + 0.160T + 23T^{2} \)
29 \( 1 + (-2.70 - 2.70i)T + 29iT^{2} \)
31 \( 1 + 4.64T + 31T^{2} \)
37 \( 1 + (5.35 + 5.35i)T + 37iT^{2} \)
41 \( 1 + 9.89iT - 41T^{2} \)
43 \( 1 + (-7.23 - 7.23i)T + 43iT^{2} \)
47 \( 1 + 4.79iT - 47T^{2} \)
53 \( 1 + (3.44 + 3.44i)T + 53iT^{2} \)
59 \( 1 + (-0.101 + 0.101i)T - 59iT^{2} \)
61 \( 1 + (6.01 + 6.01i)T + 61iT^{2} \)
67 \( 1 + (9.04 - 9.04i)T - 67iT^{2} \)
71 \( 1 + 4.60iT - 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + 5.73T + 79T^{2} \)
83 \( 1 + (2.04 - 2.04i)T - 83iT^{2} \)
89 \( 1 + 15.0iT - 89T^{2} \)
97 \( 1 + 3.84iT - 97T^{2} \)
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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

−12.41712386161085721264877440345, −11.58800986204876359383545440459, −10.12394712975701035124942617068, −9.109213181182375994542983783829, −7.76262584785059934946352110836, −7.14100488202483894935303544939, −5.68699427031778291723456899339, −4.72665672581294174822684916701, −3.46518824452045437091568028452, −1.90520791722310679332018252849, 2.71864489357790385940578767572, 3.33797140220774806870535730308, 4.80928061559200491340764855500, 5.94941532002498377097074796060, 7.13671902048191007170945538366, 7.959835555798791754890422549141, 9.620380904755338028696343009750, 10.51963599535173958687874067270, 11.21187274027475235757744931743, 12.26804002753464631514889519085

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