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Properties

Label 2-240-60.59-c1-0-3
Degree $2$
Conductor $240$
Sign $0.968 - 0.250i$
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 + (1.73 − 1.41i)5-s + (1.00 − 2.82i)9-s + 4.89·11-s + 4.89i·13-s + (−1.03 + 3.73i)15-s + 3.46·17-s + 3.46i·19-s − 6i·23-s + (0.999 − 4.89i)25-s + (1.41 + 5.00i)27-s + 2.82i·29-s − 3.46i·31-s + (−6.92 + 4.89i)33-s − 4.89i·37-s + ⋯
L(s)  = 1  + (−0.816 + 0.577i)3-s + (0.774 − 0.632i)5-s + (0.333 − 0.942i)9-s + 1.47·11-s + 1.35i·13-s + (−0.267 + 0.963i)15-s + 0.840·17-s + 0.794i·19-s − 1.25i·23-s + (0.199 − 0.979i)25-s + (0.272 + 0.962i)27-s + 0.525i·29-s − 0.622i·31-s + (−1.20 + 0.852i)33-s − 0.805i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.968 - 0.250i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.968 - 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15677 + 0.147119i\)
\(L(\frac12)\) \(\approx\) \(1.15677 + 0.147119i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.73 + 1.41i)T \)
good7 \( 1 + 7T^{2} \)
11 \( 1 - 4.89T + 11T^{2} \)
13 \( 1 - 4.89iT - 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 3.46iT - 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + 4.89iT - 37T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 - 4.89T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 8.48T + 67T^{2} \)
71 \( 1 + 9.79T + 71T^{2} \)
73 \( 1 - 9.79iT - 73T^{2} \)
79 \( 1 - 10.3iT - 79T^{2} \)
83 \( 1 + 6iT - 83T^{2} \)
89 \( 1 + 5.65iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−12.05105238922305863520078494085, −11.35705697245430769594833878692, −10.06203273634479094458279662095, −9.470761304722275295151821635955, −8.580438315334883441129694406104, −6.74081033296836789502910245016, −6.07264713205744188964764261880, −4.85315120194220802326056660786, −3.87790212734934366952010197785, −1.48988504725595801121177340730, 1.44352676811288184168506757576, 3.23530084215929063542659275680, 5.11646557689637885827581805634, 6.06540869446035540279636053306, 6.87457113694024335840718106988, 7.916456024911254506744543312511, 9.414269879175495662386867466968, 10.27215536476442613695704183387, 11.22064133721000356087276839691, 11.99989630170638126072522143395

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