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Properties

Label 2-240-15.8-c1-0-3
Degree $2$
Conductor $240$
Sign $0.865 - 0.501i$
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  + (1.70 + 0.292i)3-s + (2.12 + 0.707i)5-s + (−3 + 3i)7-s + (2.82 + i)9-s − 1.41i·11-s + (3.41 + 1.82i)15-s + (−4.24 − 4.24i)17-s − 4i·19-s + (−5.99 + 4.24i)21-s + (2.82 − 2.82i)23-s + (3.99 + 3i)25-s + (4.53 + 2.53i)27-s − 1.41·29-s + 2·31-s + (0.414 − 2.41i)33-s + ⋯
L(s)  = 1  + (0.985 + 0.169i)3-s + (0.948 + 0.316i)5-s + (−1.13 + 1.13i)7-s + (0.942 + 0.333i)9-s − 0.426i·11-s + (0.881 + 0.472i)15-s + (−1.02 − 1.02i)17-s − 0.917i·19-s + (−1.30 + 0.925i)21-s + (0.589 − 0.589i)23-s + (0.799 + 0.600i)25-s + (0.872 + 0.487i)27-s − 0.262·29-s + 0.359·31-s + (0.0721 − 0.420i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.865 - 0.501i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.865 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66773 + 0.448614i\)
\(L(\frac12)\) \(\approx\) \(1.66773 + 0.448614i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.292i)T \)
5 \( 1 + (-2.12 - 0.707i)T \)
good7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 + 1.41iT - 11T^{2} \)
13 \( 1 + 13iT^{2} \)
17 \( 1 + (4.24 + 4.24i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 + 1.41T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (2 - 2i)T - 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (-2 - 2i)T + 43iT^{2} \)
47 \( 1 + (5.65 + 5.65i)T + 47iT^{2} \)
53 \( 1 + (8.48 - 8.48i)T - 53iT^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (4 - 4i)T - 67iT^{2} \)
71 \( 1 - 2.82iT - 71T^{2} \)
73 \( 1 + (-3 - 3i)T + 73iT^{2} \)
79 \( 1 + 10iT - 79T^{2} \)
83 \( 1 + (2.82 - 2.82i)T - 83iT^{2} \)
89 \( 1 - 2.82T + 89T^{2} \)
97 \( 1 + (13 - 13i)T - 97iT^{2} \)
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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

−12.49725180053172871156787694619, −11.11927672544818574050156177964, −9.995887036803470859565591839400, −9.152119062834680031547838752112, −8.787150936026752713198893442427, −7.08611664775820312848205456849, −6.24738053258380440352583941239, −4.92376647355467233231507017008, −3.09196850208470991893785978549, −2.40211433567042417417539153740, 1.68927163085168541664644051961, 3.27778388909946455808643958243, 4.42428302669888849318001803689, 6.20464122240049289740984145059, 7.00364859946765129234684234254, 8.176809062897239988305952638596, 9.339410583234397489483062715837, 9.892296582830355028533161067276, 10.74810297217631204133332319594, 12.59710427499636671655077438769

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