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Properties

Label 2-48e2-8.3-c0-0-0
Degree $2$
Conductor $2304$
Sign $0.707 - 0.707i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + 2i·13-s + 25-s + 2i·37-s + 49-s − 2i·61-s + 2·73-s − 2·97-s + 2i·109-s + ⋯
L(s)  = 1  + 2i·13-s + 25-s + 2i·37-s + 49-s − 2i·61-s + 2·73-s − 2·97-s + 2i·109-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :0),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.157134780\)
\(L(\frac12)\) \(\approx\) \(1.157134780\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 - T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 2iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 2iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 2iT - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−9.256138654178166197316001342571, −8.618582661939724690539161761209, −7.77869817667419196365074396252, −6.69352018234265738754400787846, −6.53062603387413204559482472777, −5.20531561412361022727792090914, −4.51324427484928795713985504553, −3.64167956491301591203916541276, −2.49320609828736547236567061822, −1.43762214026917881073060627354, 0.871239430364633848829850410997, 2.42610052844155881412945048433, 3.25576330057765085835529047619, 4.22310695014203753438899659805, 5.35516703300717602532838583971, 5.74614482141289107057794243746, 6.88036566963366407484521607269, 7.59670940442531178970512800069, 8.306522801608302811399082525584, 9.030360332210759474978608301349

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