Properties

Label 2-48e2-8.3-c2-0-34
Degree $2$
Conductor $2304$
Sign $0.707 - 0.707i$
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  + 2i·5-s + 6.92i·7-s + 6.92·11-s − 2i·13-s − 10·17-s + 20.7·19-s − 27.7i·23-s + 21·25-s − 26i·29-s + 6.92i·31-s − 13.8·35-s + 26i·37-s + 58·41-s + 48.4·43-s − 69.2i·47-s + ⋯
L(s)  = 1  + 0.400i·5-s + 0.989i·7-s + 0.629·11-s − 0.153i·13-s − 0.588·17-s + 1.09·19-s − 1.20i·23-s + 0.839·25-s − 0.896i·29-s + 0.223i·31-s − 0.395·35-s + 0.702i·37-s + 1.41·41-s + 1.12·43-s − 1.47i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(62.7794\)
Root analytic conductor: \(7.92334\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.223576312\)
\(L(\frac12)\) \(\approx\) \(2.223576312\)
\(L(2)\) 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 \( 1 - 2iT - 25T^{2} \)
7 \( 1 - 6.92iT - 49T^{2} \)
11 \( 1 - 6.92T + 121T^{2} \)
13 \( 1 + 2iT - 169T^{2} \)
17 \( 1 + 10T + 289T^{2} \)
19 \( 1 - 20.7T + 361T^{2} \)
23 \( 1 + 27.7iT - 529T^{2} \)
29 \( 1 + 26iT - 841T^{2} \)
31 \( 1 - 6.92iT - 961T^{2} \)
37 \( 1 - 26iT - 1.36e3T^{2} \)
41 \( 1 - 58T + 1.68e3T^{2} \)
43 \( 1 - 48.4T + 1.84e3T^{2} \)
47 \( 1 + 69.2iT - 2.20e3T^{2} \)
53 \( 1 - 74iT - 2.80e3T^{2} \)
59 \( 1 + 90.0T + 3.48e3T^{2} \)
61 \( 1 + 26iT - 3.72e3T^{2} \)
67 \( 1 - 6.92T + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 46T + 5.32e3T^{2} \)
79 \( 1 - 117. iT - 6.24e3T^{2} \)
83 \( 1 + 48.4T + 6.88e3T^{2} \)
89 \( 1 - 82T + 7.92e3T^{2} \)
97 \( 1 - 2T + 9.40e3T^{2} \)
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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

−9.013427668841625637550106716452, −8.230101673181972429797429912992, −7.34608833044142979241859484668, −6.51063582142869721302433146109, −5.90395211089713032119940164207, −4.97424748181450191290868539251, −4.08486040669370148244136438803, −2.94807765796001552132122540498, −2.29457791482618979114251267847, −0.894291396568420371438610688122, 0.73248850573389047262912518206, 1.57628357372537341357225596994, 3.01108482372970073198034129232, 3.92708603298862517069102243335, 4.62990701588148983351473568654, 5.54890409103611305397396946473, 6.46207339599486713950923693547, 7.34793030861823519584973759845, 7.72340527551436006193855578515, 9.026190828816747002452542126225

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