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Properties

Label 2-3840-60.47-c0-0-1
Degree $2$
Conductor $3840$
Sign $-0.525 - 0.850i$
Analytic cond. $1.91640$
Root an. cond. $1.38434$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−1 + i)7-s − 1.00i·9-s + 1.41·11-s − 1.00·15-s − 1.41i·21-s + 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41·29-s + 2i·31-s + (−1.00 + 1.00i)33-s − 1.41·35-s + (0.707 − 0.707i)45-s − i·49-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + (0.707 + 0.707i)5-s + (−1 + i)7-s − 1.00i·9-s + 1.41·11-s − 1.00·15-s − 1.41i·21-s + 1.00i·25-s + (0.707 + 0.707i)27-s + 1.41·29-s + 2i·31-s + (−1.00 + 1.00i)33-s − 1.41·35-s + (0.707 − 0.707i)45-s − i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $-0.525 - 0.850i$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (767, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.049342516\)
\(L(\frac12)\) \(\approx\) \(1.049342516\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (1 - i)T - iT^{2} \)
11 \( 1 - 1.41T + T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1 - i)T - iT^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−9.036355179053238494146492670375, −8.567028179050148826625836275312, −6.88423211119388348815977619253, −6.67611372517904895493517297027, −5.98296947007311282626318227127, −5.36666523351654791701013101223, −4.38930324200167017740298440733, −3.37260501868786465966310428448, −2.83834540598223457255251747146, −1.44962416317147434205457569449, 0.71726285832060724189576117864, 1.53845581928166252501954961494, 2.72916692512216691360611965048, 4.03064347551291914590175457524, 4.55564719869795247139301794331, 5.72115192245080830620559876807, 6.23878698680512864672170059301, 6.78888396317214201796209711148, 7.48690461028808060385452155045, 8.423657925685737771910314932554

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