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Properties

Label 2-240-80.69-c1-0-4
Degree $2$
Conductor $240$
Sign $-0.729 - 0.683i$
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  + (0.382 + 1.36i)2-s + (0.707 + 0.707i)3-s + (−1.70 + 1.04i)4-s + (1.75 + 1.38i)5-s + (−0.692 + 1.23i)6-s − 4.66·7-s + (−2.07 − 1.92i)8-s + 1.00i·9-s + (−1.21 + 2.91i)10-s + (1.23 + 1.23i)11-s + (−1.94 − 0.471i)12-s + (4.12 + 4.12i)13-s + (−1.78 − 6.34i)14-s + (0.258 + 2.22i)15-s + (1.83 − 3.55i)16-s − 3.20i·17-s + ⋯
L(s)  = 1  + (0.270 + 0.962i)2-s + (0.408 + 0.408i)3-s + (−0.853 + 0.520i)4-s + (0.784 + 0.620i)5-s + (−0.282 + 0.503i)6-s − 1.76·7-s + (−0.731 − 0.681i)8-s + 0.333i·9-s + (−0.385 + 0.922i)10-s + (0.372 + 0.372i)11-s + (−0.561 − 0.136i)12-s + (1.14 + 1.14i)13-s + (−0.476 − 1.69i)14-s + (0.0666 + 0.573i)15-s + (0.458 − 0.888i)16-s − 0.778i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.729 - 0.683i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ -0.729 - 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.522112 + 1.32069i\)
\(L(\frac12)\) \(\approx\) \(0.522112 + 1.32069i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 - 1.36i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-1.75 - 1.38i)T \)
good7 \( 1 + 4.66T + 7T^{2} \)
11 \( 1 + (-1.23 - 1.23i)T + 11iT^{2} \)
13 \( 1 + (-4.12 - 4.12i)T + 13iT^{2} \)
17 \( 1 + 3.20iT - 17T^{2} \)
19 \( 1 + (-3.73 + 3.73i)T - 19iT^{2} \)
23 \( 1 + 0.714T + 23T^{2} \)
29 \( 1 + (1.24 - 1.24i)T - 29iT^{2} \)
31 \( 1 - 3.84T + 31T^{2} \)
37 \( 1 + (2.33 - 2.33i)T - 37iT^{2} \)
41 \( 1 + 6.81iT - 41T^{2} \)
43 \( 1 + (-1.31 + 1.31i)T - 43iT^{2} \)
47 \( 1 + 1.18iT - 47T^{2} \)
53 \( 1 + (-9.35 + 9.35i)T - 53iT^{2} \)
59 \( 1 + (6.22 + 6.22i)T + 59iT^{2} \)
61 \( 1 + (4.44 - 4.44i)T - 61iT^{2} \)
67 \( 1 + (-6.37 - 6.37i)T + 67iT^{2} \)
71 \( 1 - 6.23iT - 71T^{2} \)
73 \( 1 - 5.34T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + (4.88 + 4.88i)T + 83iT^{2} \)
89 \( 1 - 2.20iT - 89T^{2} \)
97 \( 1 + 7.39iT - 97T^{2} \)
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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

−12.94122037171871176587676852398, −11.63968068607042673563407907456, −10.11958938698386264640447131054, −9.413376300632536184412315558226, −8.868565164858692976745262084327, −7.03180752532306283398779205979, −6.64858598282116430164180757598, −5.52596207250761262261597786370, −3.95274272545986640381280863811, −2.94710714593170466111594226004, 1.14478664247644276198831144641, 2.91718750836869904099025194383, 3.81414993716924951271541222791, 5.75951897862717628725748025911, 6.22545748176132950379456518837, 8.185102534148060695266855851041, 9.125855004293555478240470883458, 9.856890529238832788107646995330, 10.64187335066490950179475608639, 12.16394365911387166919054313034

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