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Properties

Label 2-48e2-24.5-c2-0-38
Degree $2$
Conductor $2304$
Sign $0.985 - 0.169i$
Analytic cond. $62.7794$
Root an. cond. $7.92334$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.87·5-s + 11.7·7-s − 9.75·11-s − 22.6i·13-s + 22.1i·17-s + 17.7i·19-s + 14.1i·23-s − 1.20·25-s + 20.0·29-s + 39.7·31-s + 57.5·35-s + 2.40i·37-s + 64.3i·41-s + 3.19i·43-s − 41.8i·47-s + ⋯
L(s)  = 1  + 0.975·5-s + 1.68·7-s − 0.886·11-s − 1.74i·13-s + 1.30i·17-s + 0.936i·19-s + 0.614i·23-s − 0.0480·25-s + 0.689·29-s + 1.28·31-s + 1.64·35-s + 0.0649i·37-s + 1.56i·41-s + 0.0742i·43-s − 0.890i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.985 - 0.169i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ 0.985 - 0.169i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.156721191\)
\(L(\frac12)\) \(\approx\) \(3.156721191\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 - 4.87T + 25T^{2} \)
7 \( 1 - 11.7T + 49T^{2} \)
11 \( 1 + 9.75T + 121T^{2} \)
13 \( 1 + 22.6iT - 169T^{2} \)
17 \( 1 - 22.1iT - 289T^{2} \)
19 \( 1 - 17.7iT - 361T^{2} \)
23 \( 1 - 14.1iT - 529T^{2} \)
29 \( 1 - 20.0T + 841T^{2} \)
31 \( 1 - 39.7T + 961T^{2} \)
37 \( 1 - 2.40iT - 1.36e3T^{2} \)
41 \( 1 - 64.3iT - 1.68e3T^{2} \)
43 \( 1 - 3.19iT - 1.84e3T^{2} \)
47 \( 1 + 41.8iT - 2.20e3T^{2} \)
53 \( 1 - 55.5T + 2.80e3T^{2} \)
59 \( 1 - 111.T + 3.48e3T^{2} \)
61 \( 1 + 10.8iT - 3.72e3T^{2} \)
67 \( 1 - 18.2iT - 4.48e3T^{2} \)
71 \( 1 - 34.7iT - 5.04e3T^{2} \)
73 \( 1 + 87.5T + 5.32e3T^{2} \)
79 \( 1 - 151.T + 6.24e3T^{2} \)
83 \( 1 - 61.8T + 6.88e3T^{2} \)
89 \( 1 + 72.9iT - 7.92e3T^{2} \)
97 \( 1 + 87.3T + 9.40e3T^{2} \)
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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

−8.440531230864072711189844871398, −8.209981495667367474822879773393, −7.61112989049503751241637740007, −6.29679921905575061898057046220, −5.50776472838328620410810200578, −5.17954468705357251411274196430, −4.07162557074032176594396634565, −2.84108817724868967769985532177, −1.91579717272191126787155986084, −1.04767241444203515757892560820, 0.900393969462586240041323257350, 2.11819442129095730662223362266, 2.51065490758363473467761325002, 4.26190885303015360374419600085, 4.88121392064914742930694302026, 5.44056868063344999948424780995, 6.57002513857968924628865023627, 7.21353232398328849396597660202, 8.106008246510556237859375315004, 8.856862260793305792321016866200

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