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Properties

Label 2-3840-120.29-c0-0-6
Degree $2$
Conductor $3840$
Sign $-1$
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 − 5-s − 1.41i·7-s − 1.00i·9-s + (0.707 − 0.707i)15-s + (1.00 + 1.00i)21-s − 1.41·23-s + 25-s + (0.707 + 0.707i)27-s + 1.41i·35-s + 2i·41-s − 1.41·43-s + 1.00i·45-s − 1.41·47-s − 1.00·49-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s − 5-s − 1.41i·7-s − 1.00i·9-s + (0.707 − 0.707i)15-s + (1.00 + 1.00i)21-s − 1.41·23-s + 25-s + (0.707 + 0.707i)27-s + 1.41i·35-s + 2i·41-s − 1.41·43-s + 1.00i·45-s − 1.41·47-s − 1.00·49-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $-1$
Analytic conductor: \(1.91640\)
Root analytic conductor: \(1.38434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3840,\ (\ :0),\ -1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01237978921\)
\(L(\frac12)\) \(\approx\) \(0.01237978921\)
\(L(1)\) 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 \( 1 + T \)
good7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 + 1.41T + T^{2} \)
47 \( 1 + 1.41T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + 1.41iT - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.121505881133523541920206691540, −7.64082764386813457700430676262, −6.73260515218850031697542775123, −6.22461103655919537040496537999, −5.01200135443536793589265335769, −4.45057210495473343033195865516, −3.81356137558899594991905410199, −3.15671669102598803735436745692, −1.31464935370800166507967300460, −0.008150018827572329029354412215, 1.68649776864335353221787410115, 2.57070507008022208528793041421, 3.63240466849811611118150125263, 4.67052556955220412421916382772, 5.40954698290835439253901618813, 6.06118736793739334712054466473, 6.81324003973230557064941629380, 7.57192710449211278444612557541, 8.305461805018408531215567204887, 8.697286999566838953986846959090

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