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Properties

Label 2-240-16.13-c1-0-11
Degree $2$
Conductor $240$
Sign $0.984 + 0.177i$
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  + (−1.13 + 0.847i)2-s + (0.707 + 0.707i)3-s + (0.563 − 1.91i)4-s + (0.707 − 0.707i)5-s + (−1.39 − 0.201i)6-s − 4.27i·7-s + (0.989 + 2.64i)8-s + 1.00i·9-s + (−0.201 + 1.39i)10-s + (2.94 − 2.94i)11-s + (1.75 − 0.958i)12-s + (−4.05 − 4.05i)13-s + (3.62 + 4.83i)14-s + 1.00·15-s + (−3.36 − 2.16i)16-s + 0.160·17-s + ⋯
L(s)  = 1  + (−0.800 + 0.599i)2-s + (0.408 + 0.408i)3-s + (0.281 − 0.959i)4-s + (0.316 − 0.316i)5-s + (−0.571 − 0.0821i)6-s − 1.61i·7-s + (0.349 + 0.936i)8-s + 0.333i·9-s + (−0.0636 + 0.442i)10-s + (0.887 − 0.887i)11-s + (0.506 − 0.276i)12-s + (−1.12 − 1.12i)13-s + (0.967 + 1.29i)14-s + 0.258·15-s + (−0.841 − 0.540i)16-s + 0.0388·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 + 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.984 + 0.177i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.984 + 0.177i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00795 - 0.0900325i\)
\(L(\frac12)\) \(\approx\) \(1.00795 - 0.0900325i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 0.847i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 + 4.27iT - 7T^{2} \)
11 \( 1 + (-2.94 + 2.94i)T - 11iT^{2} \)
13 \( 1 + (4.05 + 4.05i)T + 13iT^{2} \)
17 \( 1 - 0.160T + 17T^{2} \)
19 \( 1 + (-4.32 - 4.32i)T + 19iT^{2} \)
23 \( 1 - 8.40iT - 23T^{2} \)
29 \( 1 + (1.78 + 1.78i)T + 29iT^{2} \)
31 \( 1 - 7.17T + 31T^{2} \)
37 \( 1 + (0.669 - 0.669i)T - 37iT^{2} \)
41 \( 1 - 3.96iT - 41T^{2} \)
43 \( 1 + (0.255 - 0.255i)T - 43iT^{2} \)
47 \( 1 - 0.0752T + 47T^{2} \)
53 \( 1 + (2.88 - 2.88i)T - 53iT^{2} \)
59 \( 1 + (5.63 - 5.63i)T - 59iT^{2} \)
61 \( 1 + (4.48 + 4.48i)T + 61iT^{2} \)
67 \( 1 + (0.131 + 0.131i)T + 67iT^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 0.382iT - 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 + (5.54 + 5.54i)T + 83iT^{2} \)
89 \( 1 - 13.8iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−11.85985239257556447826169469356, −10.73081365025886425095119712603, −9.932634166999336917723454699693, −9.381908713744275402906523587750, −7.960308303691833554698443469856, −7.51119500017597123912899865931, −6.11380662704711512290748065881, −4.93396000337896008576185696147, −3.44237075756310115590585017018, −1.13432239849526866074295216074, 2.00349366548623411071674463114, 2.77943473221641575026161701356, 4.65397508232973392412560533494, 6.49105109231236456136267837357, 7.22317289870293016838503490521, 8.606559891349605344038956058998, 9.267149831493328057929314973237, 9.891789026742433817662867901489, 11.41652412433458264667060987481, 12.12575994105587085356531068546

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