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Properties

Label 2-240-80.27-c1-0-19
Degree $2$
Conductor $240$
Sign $0.474 + 0.880i$
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  + (1.20 − 0.733i)2-s + 3-s + (0.924 − 1.77i)4-s + (−2.15 − 0.609i)5-s + (1.20 − 0.733i)6-s + (0.566 + 0.566i)7-s + (−0.181 − 2.82i)8-s + 9-s + (−3.04 + 0.840i)10-s + (3.64 − 3.64i)11-s + (0.924 − 1.77i)12-s + 2.74i·13-s + (1.10 + 0.269i)14-s + (−2.15 − 0.609i)15-s + (−2.28 − 3.28i)16-s + (2.08 + 2.08i)17-s + ⋯
L(s)  = 1  + (0.855 − 0.518i)2-s + 0.577·3-s + (0.462 − 0.886i)4-s + (−0.962 − 0.272i)5-s + (0.493 − 0.299i)6-s + (0.214 + 0.214i)7-s + (−0.0642 − 0.997i)8-s + 0.333·9-s + (−0.964 + 0.265i)10-s + (1.09 − 1.09i)11-s + (0.267 − 0.511i)12-s + 0.760i·13-s + (0.294 + 0.0721i)14-s + (−0.555 − 0.157i)15-s + (−0.572 − 0.820i)16-s + (0.505 + 0.505i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.474 + 0.880i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.474 + 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.79665 - 1.07296i\)
\(L(\frac12)\) \(\approx\) \(1.79665 - 1.07296i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.20 + 0.733i)T \)
3 \( 1 - T \)
5 \( 1 + (2.15 + 0.609i)T \)
good7 \( 1 + (-0.566 - 0.566i)T + 7iT^{2} \)
11 \( 1 + (-3.64 + 3.64i)T - 11iT^{2} \)
13 \( 1 - 2.74iT - 13T^{2} \)
17 \( 1 + (-2.08 - 2.08i)T + 17iT^{2} \)
19 \( 1 + (5.79 - 5.79i)T - 19iT^{2} \)
23 \( 1 + (4.28 - 4.28i)T - 23iT^{2} \)
29 \( 1 + (2.63 + 2.63i)T + 29iT^{2} \)
31 \( 1 - 8.10iT - 31T^{2} \)
37 \( 1 + 2.28iT - 37T^{2} \)
41 \( 1 + 2.27iT - 41T^{2} \)
43 \( 1 + 3.06iT - 43T^{2} \)
47 \( 1 + (-1.80 + 1.80i)T - 47iT^{2} \)
53 \( 1 - 6.32T + 53T^{2} \)
59 \( 1 + (5.56 + 5.56i)T + 59iT^{2} \)
61 \( 1 + (-4.82 + 4.82i)T - 61iT^{2} \)
67 \( 1 - 3.34iT - 67T^{2} \)
71 \( 1 + 2.81T + 71T^{2} \)
73 \( 1 + (-10.7 - 10.7i)T + 73iT^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + 1.97T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 + (1.02 + 1.02i)T + 97iT^{2} \)
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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

−12.00319049270623813607908411811, −11.33891124898287851103162258908, −10.25229122634434576755368434862, −8.982275788574276984064636241722, −8.164287813352658027465965596063, −6.78821567145435689455222543818, −5.65269915901481498528479726604, −4.04121773829139574727379628272, −3.62434216172141681033870091683, −1.69500213047663931395373284405, 2.58285792517752135436710432922, 3.97796958605964130647705380411, 4.63460141381413886454986577104, 6.40199995912672957177954498942, 7.30549113584791895919042508402, 8.038941439345632543862791958844, 9.126121636647461984959116969995, 10.58681005151296999827977713672, 11.63283657733915805359325980220, 12.41353617093473533684857699175

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