Properties

Label 2-240-1.1-c1-0-1
Degree $2$
Conductor $240$
Sign $1$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s + 4·11-s + 6·13-s − 15-s − 6·17-s + 4·19-s + 25-s − 27-s − 2·29-s + 8·31-s − 4·33-s − 2·37-s − 6·39-s − 6·41-s − 12·43-s + 45-s − 8·47-s − 7·49-s + 6·51-s + 6·53-s + 4·55-s − 4·57-s − 12·59-s + 14·61-s + 6·65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 1.66·13-s − 0.258·15-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s − 0.328·37-s − 0.960·39-s − 0.937·41-s − 1.82·43-s + 0.149·45-s − 1.16·47-s − 49-s + 0.840·51-s + 0.824·53-s + 0.539·55-s − 0.529·57-s − 1.56·59-s + 1.79·61-s + 0.744·65-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(240\)    =    \(2^{4} \cdot 3 \cdot 5\)
Sign: $1$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{240} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 240,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.206444996\)
\(L(\frac12)\) \(\approx\) \(1.206444996\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−11.83885673495833444158744201307, −11.34643695531064671651709069219, −10.28908580294693488189641431266, −9.234498555197333169762430563636, −8.372242663067272040741337380641, −6.71560882928791827542770443897, −6.23234003008114203357336027651, −4.86312932086210572949074191291, −3.57148112685573562185271725821, −1.48909737355432343830877670544, 1.48909737355432343830877670544, 3.57148112685573562185271725821, 4.86312932086210572949074191291, 6.23234003008114203357336027651, 6.71560882928791827542770443897, 8.372242663067272040741337380641, 9.234498555197333169762430563636, 10.28908580294693488189641431266, 11.34643695531064671651709069219, 11.83885673495833444158744201307

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