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Properties

Label 2-240-80.29-c1-0-11
Degree $2$
Conductor $240$
Sign $0.972 - 0.233i$
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  + (1.36 − 0.386i)2-s + (−0.707 + 0.707i)3-s + (1.70 − 1.05i)4-s + (−1.03 + 1.98i)5-s + (−0.688 + 1.23i)6-s + 3.91·7-s + (1.90 − 2.08i)8-s − 1.00i·9-s + (−0.635 + 3.09i)10-s + (−2.93 + 2.93i)11-s + (−0.459 + 1.94i)12-s + (0.732 − 0.732i)13-s + (5.33 − 1.51i)14-s + (−0.674 − 2.13i)15-s + (1.78 − 3.57i)16-s − 2.89i·17-s + ⋯
L(s)  = 1  + (0.961 − 0.273i)2-s + (−0.408 + 0.408i)3-s + (0.850 − 0.525i)4-s + (−0.461 + 0.887i)5-s + (−0.281 + 0.504i)6-s + 1.48·7-s + (0.674 − 0.738i)8-s − 0.333i·9-s + (−0.201 + 0.979i)10-s + (−0.884 + 0.884i)11-s + (−0.132 + 0.561i)12-s + (0.203 − 0.203i)13-s + (1.42 − 0.404i)14-s + (−0.174 − 0.550i)15-s + (0.447 − 0.894i)16-s − 0.701i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.972 - 0.233i$
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: \((2,\ 240,\ (\ :1/2),\ 0.972 - 0.233i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94229 + 0.229571i\)
\(L(\frac12)\) \(\approx\) \(1.94229 + 0.229571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.386i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (1.03 - 1.98i)T \)
good7 \( 1 - 3.91T + 7T^{2} \)
11 \( 1 + (2.93 - 2.93i)T - 11iT^{2} \)
13 \( 1 + (-0.732 + 0.732i)T - 13iT^{2} \)
17 \( 1 + 2.89iT - 17T^{2} \)
19 \( 1 + (-1.67 - 1.67i)T + 19iT^{2} \)
23 \( 1 - 1.73T + 23T^{2} \)
29 \( 1 + (4.99 + 4.99i)T + 29iT^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 + (6.41 + 6.41i)T + 37iT^{2} \)
41 \( 1 - 0.00577iT - 41T^{2} \)
43 \( 1 + (-2.23 - 2.23i)T + 43iT^{2} \)
47 \( 1 - 11.6iT - 47T^{2} \)
53 \( 1 + (5.55 + 5.55i)T + 53iT^{2} \)
59 \( 1 + (-3.83 + 3.83i)T - 59iT^{2} \)
61 \( 1 + (-9.30 - 9.30i)T + 61iT^{2} \)
67 \( 1 + (-3.85 + 3.85i)T - 67iT^{2} \)
71 \( 1 - 1.15iT - 71T^{2} \)
73 \( 1 - 7.98T + 73T^{2} \)
79 \( 1 - 0.843T + 79T^{2} \)
83 \( 1 + (-5.20 + 5.20i)T - 83iT^{2} \)
89 \( 1 + 5.40iT - 89T^{2} \)
97 \( 1 - 2.24iT - 97T^{2} \)
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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

−12.00454062067384333252379401191, −11.11488310412490721980215153089, −10.82222236492134156286977150954, −9.665223833108853486452858510082, −7.79246571509301384067234415038, −7.18353521683225614922531487394, −5.64118061833551842205664614934, −4.84098489546058006585355675188, −3.71144116318519267673970211669, −2.17038001498470917043736026641, 1.71449996771930991339079456979, 3.72653498932564409733962072746, 5.13642185922343618497307806931, 5.43746810889898005130742063555, 7.10049271665277240798740137925, 8.021372560074186781826489329063, 8.677616630970274340377618064996, 10.88530203444071965069562701253, 11.23856703120401165816644833020, 12.21391623172587334323953479579

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