Properties

Label 2-240-5.4-c7-0-0
Degree $2$
Conductor $240$
Sign $-0.376 - 0.926i$
Analytic cond. $74.9724$
Root an. cond. $8.65866$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 27i·3-s + (258. − 105. i)5-s − 523. i·7-s − 729·9-s − 5.22e3·11-s − 1.22e4i·13-s + (−2.84e3 − 6.99e3i)15-s + 3.23e4i·17-s − 1.72e4·19-s − 1.41e4·21-s + 9.39e4i·23-s + (5.59e4 − 5.44e4i)25-s + 1.96e4i·27-s − 2.12e5·29-s − 1.87e5·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.926 − 0.376i)5-s − 0.576i·7-s − 0.333·9-s − 1.18·11-s − 1.54i·13-s + (−0.217 − 0.534i)15-s + 1.59i·17-s − 0.575·19-s − 0.333·21-s + 1.60i·23-s + (0.716 − 0.697i)25-s + 0.192i·27-s − 1.61·29-s − 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.376 - 0.926i$
Analytic conductor: \(74.9724\)
Root analytic conductor: \(8.65866\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :7/2),\ -0.376 - 0.926i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.1388020839\)
\(L(\frac12)\) \(\approx\) \(0.1388020839\)
\(L(\frac{9}{2})\) 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 \( 1 + 27iT \)
5 \( 1 + (-258. + 105. i)T \)
good7 \( 1 + 523. iT - 8.23e5T^{2} \)
11 \( 1 + 5.22e3T + 1.94e7T^{2} \)
13 \( 1 + 1.22e4iT - 6.27e7T^{2} \)
17 \( 1 - 3.23e4iT - 4.10e8T^{2} \)
19 \( 1 + 1.72e4T + 8.93e8T^{2} \)
23 \( 1 - 9.39e4iT - 3.40e9T^{2} \)
29 \( 1 + 2.12e5T + 1.72e10T^{2} \)
31 \( 1 + 1.87e5T + 2.75e10T^{2} \)
37 \( 1 - 4.38e4iT - 9.49e10T^{2} \)
41 \( 1 - 7.67e5T + 1.94e11T^{2} \)
43 \( 1 + 4.18e5iT - 2.71e11T^{2} \)
47 \( 1 + 6.98e5iT - 5.06e11T^{2} \)
53 \( 1 - 1.03e6iT - 1.17e12T^{2} \)
59 \( 1 + 2.58e6T + 2.48e12T^{2} \)
61 \( 1 - 9.28e5T + 3.14e12T^{2} \)
67 \( 1 - 7.07e5iT - 6.06e12T^{2} \)
71 \( 1 + 3.59e6T + 9.09e12T^{2} \)
73 \( 1 - 3.64e6iT - 1.10e13T^{2} \)
79 \( 1 - 4.35e6T + 1.92e13T^{2} \)
83 \( 1 - 4.27e6iT - 2.71e13T^{2} \)
89 \( 1 - 7.49e6T + 4.42e13T^{2} \)
97 \( 1 - 3.96e5iT - 8.07e13T^{2} \)
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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

−10.83680958344000759170943747096, −10.43203690432144375085236115698, −9.233005065684394223467691370842, −8.064987696082940607451695990391, −7.39466584631331800361783691029, −5.85025910775628845455683033327, −5.44641850738762867274402649915, −3.72009463984549682661281604667, −2.32623910805322252251710652620, −1.26193519245861977006390298922, 0.03030873795636829214411695287, 2.05639855776435127798465020762, 2.79535793367946781958658679074, 4.45071430512726095716393080535, 5.40775700006163893198999588953, 6.40177484219752527329527235593, 7.54714064365080301939730781968, 9.105769606208334923911037996322, 9.384707551652642988295225687168, 10.63583600944469076802765886306

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