Properties

Label 2-240-80.29-c1-0-10
Degree $2$
Conductor $240$
Sign $0.642 + 0.766i$
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  + (0.456 − 1.33i)2-s + (−0.707 + 0.707i)3-s + (−1.58 − 1.22i)4-s + (1.65 + 1.50i)5-s + (0.623 + 1.26i)6-s + 2.58·7-s + (−2.35 + 1.55i)8-s − 1.00i·9-s + (2.76 − 1.53i)10-s + (4.39 − 4.39i)11-s + (1.98 − 0.254i)12-s + (0.417 − 0.417i)13-s + (1.18 − 3.46i)14-s + (−2.23 + 0.110i)15-s + (1.00 + 3.87i)16-s + 4.40i·17-s + ⋯
L(s)  = 1  + (0.323 − 0.946i)2-s + (−0.408 + 0.408i)3-s + (−0.791 − 0.611i)4-s + (0.741 + 0.671i)5-s + (0.254 + 0.518i)6-s + 0.978·7-s + (−0.834 + 0.551i)8-s − 0.333i·9-s + (0.874 − 0.484i)10-s + (1.32 − 1.32i)11-s + (0.572 − 0.0734i)12-s + (0.115 − 0.115i)13-s + (0.316 − 0.926i)14-s + (−0.576 + 0.0286i)15-s + (0.252 + 0.967i)16-s + 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{240} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 0.642 + 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31791 - 0.615234i\)
\(L(\frac12)\) \(\approx\) \(1.31791 - 0.615234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.456 + 1.33i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-1.65 - 1.50i)T \)
good7 \( 1 - 2.58T + 7T^{2} \)
11 \( 1 + (-4.39 + 4.39i)T - 11iT^{2} \)
13 \( 1 + (-0.417 + 0.417i)T - 13iT^{2} \)
17 \( 1 - 4.40iT - 17T^{2} \)
19 \( 1 + (4.53 + 4.53i)T + 19iT^{2} \)
23 \( 1 - 0.281T + 23T^{2} \)
29 \( 1 + (-3.73 - 3.73i)T + 29iT^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 + (5.26 + 5.26i)T + 37iT^{2} \)
41 \( 1 - 5.16iT - 41T^{2} \)
43 \( 1 + (2.66 + 2.66i)T + 43iT^{2} \)
47 \( 1 - 7.45iT - 47T^{2} \)
53 \( 1 + (2.89 + 2.89i)T + 53iT^{2} \)
59 \( 1 + (4.60 - 4.60i)T - 59iT^{2} \)
61 \( 1 + (0.211 + 0.211i)T + 61iT^{2} \)
67 \( 1 + (7.17 - 7.17i)T - 67iT^{2} \)
71 \( 1 + 15.9iT - 71T^{2} \)
73 \( 1 + 10.5T + 73T^{2} \)
79 \( 1 - 4.53T + 79T^{2} \)
83 \( 1 + (4.56 - 4.56i)T - 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + 6.78iT - 97T^{2} \)
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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

−11.69245035650146823636866448403, −10.92259079693704072243046512847, −10.59362878013374763064707020657, −9.220233609711683303320128178013, −8.545616129177440884995955333839, −6.52036187673477654342208087112, −5.71606955953733953093652752652, −4.46131006306247884532694875343, −3.26215502631623561500228519336, −1.57845476049274832266405837581, 1.71438705236091204912060817954, 4.30744080537319655180035187289, 5.05214612703059771120254200805, 6.21474690329066604446733738917, 7.08368858609646375468937797917, 8.223369151266514383190173178355, 9.131165840007377090147210201494, 10.11998303382172906792178205048, 11.81245760433474292264619303888, 12.27073962227662985436977454576

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