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Properties

Label 2-240-16.5-c1-0-0
Degree $2$
Conductor $240$
Sign $-0.346 - 0.937i$
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  + (−1.38 − 0.289i)2-s + (−0.707 + 0.707i)3-s + (1.83 + 0.800i)4-s + (−0.707 − 0.707i)5-s + (1.18 − 0.774i)6-s + 2.60i·7-s + (−2.30 − 1.63i)8-s − 1.00i·9-s + (0.774 + 1.18i)10-s + (0.702 + 0.702i)11-s + (−1.86 + 0.729i)12-s + (−2.12 + 2.12i)13-s + (0.754 − 3.61i)14-s + 1.00·15-s + (2.71 + 2.93i)16-s − 2.33·17-s + ⋯
L(s)  = 1  + (−0.978 − 0.204i)2-s + (−0.408 + 0.408i)3-s + (0.916 + 0.400i)4-s + (−0.316 − 0.316i)5-s + (0.483 − 0.316i)6-s + 0.985i·7-s + (−0.815 − 0.579i)8-s − 0.333i·9-s + (0.244 + 0.374i)10-s + (0.211 + 0.211i)11-s + (−0.537 + 0.210i)12-s + (−0.589 + 0.589i)13-s + (0.201 − 0.964i)14-s + 0.258·15-s + (0.679 + 0.733i)16-s − 0.566·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.346 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.346 - 0.937i$
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: \((2,\ 240,\ (\ :1/2),\ -0.346 - 0.937i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.267901 + 0.384670i\)
\(L(\frac12)\) \(\approx\) \(0.267901 + 0.384670i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.289i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (0.707 + 0.707i)T \)
good7 \( 1 - 2.60iT - 7T^{2} \)
11 \( 1 + (-0.702 - 0.702i)T + 11iT^{2} \)
13 \( 1 + (2.12 - 2.12i)T - 13iT^{2} \)
17 \( 1 + 2.33T + 17T^{2} \)
19 \( 1 + (3.46 - 3.46i)T - 19iT^{2} \)
23 \( 1 - 8.48iT - 23T^{2} \)
29 \( 1 + (5.11 - 5.11i)T - 29iT^{2} \)
31 \( 1 - 5.61T + 31T^{2} \)
37 \( 1 + (0.967 + 0.967i)T + 37iT^{2} \)
41 \( 1 + 3.19iT - 41T^{2} \)
43 \( 1 + (2.92 + 2.92i)T + 43iT^{2} \)
47 \( 1 - 8.59T + 47T^{2} \)
53 \( 1 + (3.45 + 3.45i)T + 53iT^{2} \)
59 \( 1 + (5.32 + 5.32i)T + 59iT^{2} \)
61 \( 1 + (-9.33 + 9.33i)T - 61iT^{2} \)
67 \( 1 + (0.273 - 0.273i)T - 67iT^{2} \)
71 \( 1 - 15.3iT - 71T^{2} \)
73 \( 1 + 8.55iT - 73T^{2} \)
79 \( 1 + 4.46T + 79T^{2} \)
83 \( 1 + (-9.85 + 9.85i)T - 83iT^{2} \)
89 \( 1 - 7.30iT - 89T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−12.01110899930651087616277547187, −11.52563496295643238384083730355, −10.43473676296878157089387399764, −9.387588319652478002802015764188, −8.836952760448390769921357104376, −7.65130046627906471944812446585, −6.49368610499294690585987113067, −5.30284884420987132908808454280, −3.73642008778375405952431799249, −1.99373405991124308239056467708, 0.52284184282425862529204421686, 2.55082613553367527819710498255, 4.48793254993903190794624879504, 6.16464808550574185754372553056, 6.95448294606938974753164853013, 7.78894372080149902840500774321, 8.771343755381806351332674795343, 10.12909257016565341914577764878, 10.74080618858653431676534921746, 11.54199590211143102410818908803

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