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Properties

Label 2-240-5.4-c3-0-17
Degree $2$
Conductor $240$
Sign $-0.999 + 0.0160i$
Analytic cond. $14.1604$
Root an. cond. $3.76303$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3i·3-s + (−0.178 − 11.1i)5-s − 35.0i·7-s − 9·9-s − 25.6·11-s + 37.6i·13-s + (−33.5 + 0.536i)15-s + 95.7i·17-s + 50.8·19-s − 105.·21-s − 110. i·23-s + (−124. + 4i)25-s + 27i·27-s + 54.5·29-s − 198.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (−0.0160 − 0.999i)5-s − 1.89i·7-s − 0.333·9-s − 0.702·11-s + 0.803i·13-s + (−0.577 + 0.00923i)15-s + 1.36i·17-s + 0.614·19-s − 1.09·21-s − 1.00i·23-s + (−0.999 + 0.0320i)25-s + 0.192i·27-s + 0.349·29-s − 1.14·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0160i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $-0.999 + 0.0160i$
Analytic conductor: \(14.1604\)
Root analytic conductor: \(3.76303\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :3/2),\ -0.999 + 0.0160i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.00888635 - 1.11058i\)
\(L(\frac12)\) \(\approx\) \(0.00888635 - 1.11058i\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
5 \( 1 + (0.178 + 11.1i)T \)
good7 \( 1 + 35.0iT - 343T^{2} \)
11 \( 1 + 25.6T + 1.33e3T^{2} \)
13 \( 1 - 37.6iT - 2.19e3T^{2} \)
17 \( 1 - 95.7iT - 4.91e3T^{2} \)
19 \( 1 - 50.8T + 6.85e3T^{2} \)
23 \( 1 + 110. iT - 1.21e4T^{2} \)
29 \( 1 - 54.5T + 2.43e4T^{2} \)
31 \( 1 + 198.T + 2.97e4T^{2} \)
37 \( 1 + 266. iT - 5.06e4T^{2} \)
41 \( 1 - 103.T + 6.89e4T^{2} \)
43 \( 1 + 108iT - 7.95e4T^{2} \)
47 \( 1 - 597. iT - 1.03e5T^{2} \)
53 \( 1 + 305. iT - 1.48e5T^{2} \)
59 \( 1 + 223.T + 2.05e5T^{2} \)
61 \( 1 - 485.T + 2.26e5T^{2} \)
67 \( 1 + 876. iT - 3.00e5T^{2} \)
71 \( 1 + 585.T + 3.57e5T^{2} \)
73 \( 1 + 1.13e3iT - 3.89e5T^{2} \)
79 \( 1 - 685.T + 4.93e5T^{2} \)
83 \( 1 + 305. iT - 5.71e5T^{2} \)
89 \( 1 + 887.T + 7.04e5T^{2} \)
97 \( 1 - 556. iT - 9.12e5T^{2} \)
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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

−11.11175578288673786317611238596, −10.38234440980093782442343390643, −9.195129025820703077377122719733, −8.037510290373641179944365949318, −7.36309866828901167196422079513, −6.18896796102865174884084972502, −4.74796193040842460574158391353, −3.78806937567154481421991053646, −1.67458396370741818752482300532, −0.44017551898848511826201330403, 2.49961775478193313181638303235, 3.22133505303984219599411898940, 5.20595113947110746187314628982, 5.75302322313873772384326035754, 7.21081906028568915162418290393, 8.336417974054437322523780050378, 9.405092260829070861486099379508, 10.13675687228175080325443623427, 11.35114036273896951875872514689, 11.84121038429802159321164183724

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