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Properties

Label 2-240-80.29-c1-0-21
Degree $2$
Conductor $240$
Sign $-0.365 + 0.930i$
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  + (0.903 − 1.08i)2-s + (0.707 − 0.707i)3-s + (−0.368 − 1.96i)4-s + (−2.09 − 0.770i)5-s + (−0.130 − 1.40i)6-s + 3.05·7-s + (−2.47 − 1.37i)8-s − 1.00i·9-s + (−2.73 + 1.58i)10-s + (−1.80 + 1.80i)11-s + (−1.65 − 1.12i)12-s + (2.47 − 2.47i)13-s + (2.75 − 3.31i)14-s + (−2.02 + 0.939i)15-s + (−3.72 + 1.44i)16-s + 3.66i·17-s + ⋯
L(s)  = 1  + (0.638 − 0.769i)2-s + (0.408 − 0.408i)3-s + (−0.184 − 0.982i)4-s + (−0.938 − 0.344i)5-s + (−0.0533 − 0.574i)6-s + 1.15·7-s + (−0.873 − 0.486i)8-s − 0.333i·9-s + (−0.864 + 0.502i)10-s + (−0.545 + 0.545i)11-s + (−0.476 − 0.326i)12-s + (0.685 − 0.685i)13-s + (0.736 − 0.887i)14-s + (−0.523 + 0.242i)15-s + (−0.932 + 0.361i)16-s + 0.889i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.365 + 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.365 + 0.930i$
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: \((2,\ 240,\ (\ :1/2),\ -0.365 + 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.976024 - 1.43156i\)
\(L(\frac12)\) \(\approx\) \(0.976024 - 1.43156i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.903 + 1.08i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (2.09 + 0.770i)T \)
good7 \( 1 - 3.05T + 7T^{2} \)
11 \( 1 + (1.80 - 1.80i)T - 11iT^{2} \)
13 \( 1 + (-2.47 + 2.47i)T - 13iT^{2} \)
17 \( 1 - 3.66iT - 17T^{2} \)
19 \( 1 + (2.31 + 2.31i)T + 19iT^{2} \)
23 \( 1 - 4.86T + 23T^{2} \)
29 \( 1 + (-4.74 - 4.74i)T + 29iT^{2} \)
31 \( 1 - 1.86T + 31T^{2} \)
37 \( 1 + (-5.40 - 5.40i)T + 37iT^{2} \)
41 \( 1 - 6.47iT - 41T^{2} \)
43 \( 1 + (4.19 + 4.19i)T + 43iT^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 + (9.99 + 9.99i)T + 53iT^{2} \)
59 \( 1 + (2.47 - 2.47i)T - 59iT^{2} \)
61 \( 1 + (-8.01 - 8.01i)T + 61iT^{2} \)
67 \( 1 + (8.60 - 8.60i)T - 67iT^{2} \)
71 \( 1 - 6.63iT - 71T^{2} \)
73 \( 1 - 2.70T + 73T^{2} \)
79 \( 1 + 10.9T + 79T^{2} \)
83 \( 1 + (-3.65 + 3.65i)T - 83iT^{2} \)
89 \( 1 + 13.1iT - 89T^{2} \)
97 \( 1 - 12.4iT - 97T^{2} \)
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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

−11.84658716394882707040336176945, −11.11802982608581399621232779865, −10.25805219398214168943002540560, −8.678140631209826163447355254949, −8.135612500900931751523731638082, −6.79276536490574705531224213185, −5.20459945804346849178427097814, −4.33770053857889681351192730980, −2.99633196591586141444753219431, −1.35407456916246103021795413650, 2.87726850634146647881357062433, 4.15176726387079596218784301635, 4.93534926292523436538046479635, 6.37627630542708660346337649037, 7.67310803186890821313068681626, 8.177845400631334924582452523138, 9.137371981829725124142988729355, 10.91615055967893551982415430057, 11.40561903675744475243363207060, 12.48079065955752587842522291312

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