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Properties

Label 2-240-15.2-c1-0-6
Degree $2$
Conductor $240$
Sign $0.0618 + 0.998i$
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  + (−1.41 + i)3-s + (−1 − 2i)5-s + (−0.414 − 0.414i)7-s + (1.00 − 2.82i)9-s − 4.82i·11-s + (−1.82 + 1.82i)13-s + (3.41 + 1.82i)15-s + (3.82 − 3.82i)17-s − 4.82i·19-s + (1 + 0.171i)21-s + (−1.58 − 1.58i)23-s + (−3 + 4i)25-s + (1.41 + 5.00i)27-s − 7.65·29-s + 5.65·31-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)3-s + (−0.447 − 0.894i)5-s + (−0.156 − 0.156i)7-s + (0.333 − 0.942i)9-s − 1.45i·11-s + (−0.507 + 0.507i)13-s + (0.881 + 0.472i)15-s + (0.928 − 0.928i)17-s − 1.10i·19-s + (0.218 + 0.0374i)21-s + (−0.330 − 0.330i)23-s + (−0.600 + 0.800i)25-s + (0.272 + 0.962i)27-s − 1.42·29-s + 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.0618 + 0.998i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.0618 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.501033 - 0.470938i\)
\(L(\frac12)\) \(\approx\) \(0.501033 - 0.470938i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.41 - i)T \)
5 \( 1 + (1 + 2i)T \)
good7 \( 1 + (0.414 + 0.414i)T + 7iT^{2} \)
11 \( 1 + 4.82iT - 11T^{2} \)
13 \( 1 + (1.82 - 1.82i)T - 13iT^{2} \)
17 \( 1 + (-3.82 + 3.82i)T - 17iT^{2} \)
19 \( 1 + 4.82iT - 19T^{2} \)
23 \( 1 + (1.58 + 1.58i)T + 23iT^{2} \)
29 \( 1 + 7.65T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (0.171 + 0.171i)T + 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (2.41 - 2.41i)T - 43iT^{2} \)
47 \( 1 + (6.41 - 6.41i)T - 47iT^{2} \)
53 \( 1 + (3 + 3i)T + 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + (-4.07 - 4.07i)T + 67iT^{2} \)
71 \( 1 + 6.48iT - 71T^{2} \)
73 \( 1 + (-6.65 + 6.65i)T - 73iT^{2} \)
79 \( 1 - 4.82iT - 79T^{2} \)
83 \( 1 + (-5.24 - 5.24i)T + 83iT^{2} \)
89 \( 1 - 4.34T + 89T^{2} \)
97 \( 1 + (-1 - i)T + 97iT^{2} \)
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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

−11.57864446298623278250321512176, −11.32135150666001708949962608710, −9.891017560993633487925871936998, −9.162944335905438543392380625398, −8.046532119129929370378495464367, −6.71771889078974691931238367057, −5.50555446597866406431859449225, −4.66252222966729972445850437089, −3.40156015935742667575846942575, −0.61869269536148123988064184944, 2.03791085785182109504312541004, 3.80945201115467149013523338902, 5.30956871714691693882438195892, 6.36679599390417215470791497846, 7.39269586426901635017273492407, 7.978234756586393929514733428879, 9.951563273913570208722849070977, 10.35615438429173962982686885209, 11.59350415911250702280339396890, 12.28643510206877428490172570474

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