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Properties

Label 2-240-15.8-c1-0-5
Degree $2$
Conductor $240$
Sign $0.920 + 0.391i$
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 + (2.41 − 2.41i)7-s + (1.00 − 2.82i)9-s − 0.828i·11-s + (3.82 + 3.82i)13-s + (0.585 + 3.82i)15-s + (−1.82 − 1.82i)17-s − 0.828i·19-s + (1 − 5.82i)21-s + (−4.41 + 4.41i)23-s + (−3 − 4i)25-s + (−1.41 − 5.00i)27-s + 3.65·29-s − 5.65·31-s + ⋯
L(s)  = 1  + (0.816 − 0.577i)3-s + (−0.447 + 0.894i)5-s + (0.912 − 0.912i)7-s + (0.333 − 0.942i)9-s − 0.249i·11-s + (1.06 + 1.06i)13-s + (0.151 + 0.988i)15-s + (−0.443 − 0.443i)17-s − 0.190i·19-s + (0.218 − 1.27i)21-s + (−0.920 + 0.920i)23-s + (−0.600 − 0.800i)25-s + (−0.272 − 0.962i)27-s + 0.679·29-s − 1.01·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.920 + 0.391i$
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: \((2,\ 240,\ (\ :1/2),\ 0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.57350 - 0.320393i\)
\(L(\frac12)\) \(\approx\) \(1.57350 - 0.320393i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 + 2.41i)T - 7iT^{2} \)
11 \( 1 + 0.828iT - 11T^{2} \)
13 \( 1 + (-3.82 - 3.82i)T + 13iT^{2} \)
17 \( 1 + (1.82 + 1.82i)T + 17iT^{2} \)
19 \( 1 + 0.828iT - 19T^{2} \)
23 \( 1 + (4.41 - 4.41i)T - 23iT^{2} \)
29 \( 1 - 3.65T + 29T^{2} \)
31 \( 1 + 5.65T + 31T^{2} \)
37 \( 1 + (5.82 - 5.82i)T - 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-0.414 - 0.414i)T + 43iT^{2} \)
47 \( 1 + (3.58 + 3.58i)T + 47iT^{2} \)
53 \( 1 + (3 - 3i)T - 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 0.343T + 61T^{2} \)
67 \( 1 + (10.0 - 10.0i)T - 67iT^{2} \)
71 \( 1 + 10.4iT - 71T^{2} \)
73 \( 1 + (4.65 + 4.65i)T + 73iT^{2} \)
79 \( 1 - 0.828iT - 79T^{2} \)
83 \( 1 + (3.24 - 3.24i)T - 83iT^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
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

−11.81600717657478043372226928238, −11.28425969652339685152784748576, −10.23071841965130006111054966861, −8.942119423451883496346562998454, −7.997244195133179753957741507264, −7.20766969216210668848834724070, −6.35505972097129652685468766393, −4.33397540901941998130845560873, −3.35895523252105777061499125456, −1.68602546831422200502715032942, 1.96479209301956003668767726562, 3.66441629165746133048714740147, 4.76564576655909342336346123859, 5.75181553586140405454264746128, 7.70659754796252696284891729076, 8.523482516363023760874781360212, 8.870334763244681833547895305383, 10.26506673501869402104348427280, 11.16145811603391890689671448864, 12.31415478919733115348771067532

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