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Properties

Label 2-240-16.5-c1-0-4
Degree $2$
Conductor $240$
Sign $0.275 - 0.961i$
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  + (1.32 + 0.503i)2-s + (−0.707 + 0.707i)3-s + (1.49 + 1.33i)4-s + (−0.707 − 0.707i)5-s + (−1.29 + 0.578i)6-s + 2.69i·7-s + (1.30 + 2.51i)8-s − 1.00i·9-s + (−0.578 − 1.29i)10-s + (2.72 + 2.72i)11-s + (−1.99 + 0.114i)12-s + (1.82 − 1.82i)13-s + (−1.35 + 3.56i)14-s + 1.00·15-s + (0.455 + 3.97i)16-s − 7.33·17-s + ⋯
L(s)  = 1  + (0.934 + 0.356i)2-s + (−0.408 + 0.408i)3-s + (0.746 + 0.665i)4-s + (−0.316 − 0.316i)5-s + (−0.526 + 0.236i)6-s + 1.01i·7-s + (0.460 + 0.887i)8-s − 0.333i·9-s + (−0.182 − 0.408i)10-s + (0.822 + 0.822i)11-s + (−0.576 + 0.0329i)12-s + (0.506 − 0.506i)13-s + (−0.362 + 0.951i)14-s + 0.258·15-s + (0.113 + 0.993i)16-s − 1.77·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.275 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45027 + 1.09360i\)
\(L(\frac12)\) \(\approx\) \(1.45027 + 1.09360i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 - 0.503i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 - 2.69iT - 7T^{2} \)
11 \( 1 + (-2.72 - 2.72i)T + 11iT^{2} \)
13 \( 1 + (-1.82 + 1.82i)T - 13iT^{2} \)
17 \( 1 + 7.33T + 17T^{2} \)
19 \( 1 + (-3.62 + 3.62i)T - 19iT^{2} \)
23 \( 1 + 8.95iT - 23T^{2} \)
29 \( 1 + (-2.84 + 2.84i)T - 29iT^{2} \)
31 \( 1 + 3.37T + 31T^{2} \)
37 \( 1 + (0.190 + 0.190i)T + 37iT^{2} \)
41 \( 1 + 7.67iT - 41T^{2} \)
43 \( 1 + (-7.98 - 7.98i)T + 43iT^{2} \)
47 \( 1 + 1.31T + 47T^{2} \)
53 \( 1 + (6.71 + 6.71i)T + 53iT^{2} \)
59 \( 1 + (-1.01 - 1.01i)T + 59iT^{2} \)
61 \( 1 + (-2.38 + 2.38i)T - 61iT^{2} \)
67 \( 1 + (7.22 - 7.22i)T - 67iT^{2} \)
71 \( 1 + 2.28iT - 71T^{2} \)
73 \( 1 + 1.31iT - 73T^{2} \)
79 \( 1 + 2.59T + 79T^{2} \)
83 \( 1 + (5.36 - 5.36i)T - 83iT^{2} \)
89 \( 1 - 14.8iT - 89T^{2} \)
97 \( 1 - 0.694T + 97T^{2} \)
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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

−12.38373905946237225793421524169, −11.57895900005351262077046054466, −10.80625208610793904536762983555, −9.236314971735543136375322704939, −8.450622308611692122737850968856, −6.97234101175144440746782396344, −6.12387076381136943855036783535, −4.93250180573769183531041333643, −4.15571810901119128836365860342, −2.51237251636868632925002305849, 1.42424910707087048937052727043, 3.42970319318287108872590293539, 4.32746686241613772648610361423, 5.81920203850490568481175780630, 6.73165926705394642488372322670, 7.53020671176901020993592561947, 9.161698082703820711218679164563, 10.50637780692564856547255057273, 11.29175475468335359453029393268, 11.71757235582393874881143232080

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