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Properties

Label 2-240-5.4-c3-0-13
Degree $2$
Conductor $240$
Sign $0.116 + 0.993i$
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 + (11.1 − 1.29i)5-s − 16.2i·7-s − 9·9-s + 40.2·11-s + 19.7i·13-s + (−3.89 − 33.3i)15-s − 83.0i·17-s − 48.8·19-s − 48.6·21-s − 1.61i·23-s + (121. − 28.8i)25-s + 27i·27-s + 24.5·29-s + 12.4·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + (0.993 − 0.116i)5-s − 0.875i·7-s − 0.333·9-s + 1.10·11-s + 0.422i·13-s + (−0.0670 − 0.573i)15-s − 1.18i·17-s − 0.589·19-s − 0.505·21-s − 0.0146i·23-s + (0.973 − 0.230i)25-s + 0.192i·27-s + 0.157·29-s + 0.0719·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.116 + 0.993i$
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: \((2,\ 240,\ (\ :3/2),\ 0.116 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.56333 - 1.39118i\)
\(L(\frac12)\) \(\approx\) \(1.56333 - 1.39118i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-11.1 + 1.29i)T \)
good7 \( 1 + 16.2iT - 343T^{2} \)
11 \( 1 - 40.2T + 1.33e3T^{2} \)
13 \( 1 - 19.7iT - 2.19e3T^{2} \)
17 \( 1 + 83.0iT - 4.91e3T^{2} \)
19 \( 1 + 48.8T + 6.85e3T^{2} \)
23 \( 1 + 1.61iT - 1.21e4T^{2} \)
29 \( 1 - 24.5T + 2.43e4T^{2} \)
31 \( 1 - 12.4T + 2.97e4T^{2} \)
37 \( 1 + 325. iT - 5.06e4T^{2} \)
41 \( 1 + 242.T + 6.89e4T^{2} \)
43 \( 1 + 367. iT - 7.95e4T^{2} \)
47 \( 1 + 204. iT - 1.03e5T^{2} \)
53 \( 1 + 61.5iT - 1.48e5T^{2} \)
59 \( 1 + 112.T + 2.05e5T^{2} \)
61 \( 1 - 477.T + 2.26e5T^{2} \)
67 \( 1 - 558. iT - 3.00e5T^{2} \)
71 \( 1 + 558.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 - 1.15e3T + 4.93e5T^{2} \)
83 \( 1 - 1.15e3iT - 5.71e5T^{2} \)
89 \( 1 + 96.9T + 7.04e5T^{2} \)
97 \( 1 - 1.15e3iT - 9.12e5T^{2} \)
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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

−11.53084768470093063888347520869, −10.47267261437079919637815374562, −9.479229034208714605426240774177, −8.653514044012163441448701353336, −7.15884416398857355615678534789, −6.60246198968302584698964331251, −5.33714319863991330525691852953, −3.94421473548856485499658394446, −2.21989783275900112060770487565, −0.900694592795974047348870824294, 1.70917454443674480679747490419, 3.14984399947062200599464808055, 4.62509143407575368562767966912, 5.86767483691853404448270643801, 6.49657354461984138519113280230, 8.297004309622728362340769523241, 9.091877320166065453671172788202, 9.934794480255495333382902046332, 10.79775158457104795887605504225, 11.89830622728171711348820810509

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