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Properties

Label 2-240-16.5-c1-0-13
Degree $2$
Conductor $240$
Sign $0.464 + 0.885i$
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  + (1.15 − 0.811i)2-s + (0.707 − 0.707i)3-s + (0.681 − 1.88i)4-s + (0.707 + 0.707i)5-s + (0.244 − 1.39i)6-s + 2.18i·7-s + (−0.737 − 2.73i)8-s − 1.00i·9-s + (1.39 + 0.244i)10-s + (−0.00889 − 0.00889i)11-s + (−0.847 − 1.81i)12-s + (1.72 − 1.72i)13-s + (1.77 + 2.52i)14-s + 1.00·15-s + (−3.07 − 2.56i)16-s − 5.54·17-s + ⋯
L(s)  = 1  + (0.818 − 0.574i)2-s + (0.408 − 0.408i)3-s + (0.340 − 0.940i)4-s + (0.316 + 0.316i)5-s + (0.0998 − 0.568i)6-s + 0.824i·7-s + (−0.260 − 0.965i)8-s − 0.333i·9-s + (0.440 + 0.0773i)10-s + (−0.00268 − 0.00268i)11-s + (−0.244 − 0.522i)12-s + (0.479 − 0.479i)13-s + (0.473 + 0.674i)14-s + 0.258·15-s + (−0.767 − 0.640i)16-s − 1.34·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.464 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.464 + 0.885i$
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: \((2,\ 240,\ (\ :1/2),\ 0.464 + 0.885i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83941 - 1.11258i\)
\(L(\frac12)\) \(\approx\) \(1.83941 - 1.11258i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.15 + 0.811i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 - 2.18iT - 7T^{2} \)
11 \( 1 + (0.00889 + 0.00889i)T + 11iT^{2} \)
13 \( 1 + (-1.72 + 1.72i)T - 13iT^{2} \)
17 \( 1 + 5.54T + 17T^{2} \)
19 \( 1 + (4.94 - 4.94i)T - 19iT^{2} \)
23 \( 1 - 3.01iT - 23T^{2} \)
29 \( 1 + (-3.20 + 3.20i)T - 29iT^{2} \)
31 \( 1 - 3.58T + 31T^{2} \)
37 \( 1 + (-4.97 - 4.97i)T + 37iT^{2} \)
41 \( 1 + 3.76iT - 41T^{2} \)
43 \( 1 + (-6.81 - 6.81i)T + 43iT^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + (0.932 + 0.932i)T + 53iT^{2} \)
59 \( 1 + (4.60 + 4.60i)T + 59iT^{2} \)
61 \( 1 + (-4.17 + 4.17i)T - 61iT^{2} \)
67 \( 1 + (11.0 - 11.0i)T - 67iT^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 7.12iT - 73T^{2} \)
79 \( 1 + 3.41T + 79T^{2} \)
83 \( 1 + (-5.31 + 5.31i)T - 83iT^{2} \)
89 \( 1 + 5.06iT - 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−12.09807388677131203043274273389, −11.18820625837430319090884909698, −10.22279312428305593665289031135, −9.171673786681653692084164322494, −8.081829455809398255891068156763, −6.50011652448423353154514279001, −5.91610452461258547330029531494, −4.41841343166787833974849483376, −3.00330092402967084826370529964, −1.91433732994299929798318423275, 2.49787693644802985432534486970, 4.12407145091702819556722442525, 4.70297219751087427915355279680, 6.27090882177847616010605398535, 7.08074955793840541311020610586, 8.420764540872578833366762273805, 9.083040953542304625671297502171, 10.58036974979559491090441252708, 11.30551882656515435231009889754, 12.69225816723931858581536141805

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