Properties

Label 2-48e2-1.1-c1-0-37
Degree $2$
Conductor $2304$
Sign $-1$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s − 4·13-s − 8·17-s − 25-s − 10·29-s − 12·37-s + 8·41-s − 7·49-s + 14·53-s − 12·61-s − 8·65-s + 6·73-s − 16·85-s + 16·89-s + 18·97-s − 2·101-s − 20·109-s + 16·113-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.10·13-s − 1.94·17-s − 1/5·25-s − 1.85·29-s − 1.97·37-s + 1.24·41-s − 49-s + 1.92·53-s − 1.53·61-s − 0.992·65-s + 0.702·73-s − 1.73·85-s + 1.69·89-s + 1.82·97-s − 0.199·101-s − 1.91·109-s + 1.50·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-1$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2304} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 16 T + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−8.955643677804340639523778996300, −7.74398608326178337806970095846, −7.04532327351522358604131570139, −6.28655854302767590234892421883, −5.45095421797933849120346272823, −4.71660698363879354831850755374, −3.73220900163596549184218440331, −2.41954791177205934755391933624, −1.86735723343660348363185523771, 0, 1.86735723343660348363185523771, 2.41954791177205934755391933624, 3.73220900163596549184218440331, 4.71660698363879354831850755374, 5.45095421797933849120346272823, 6.28655854302767590234892421883, 7.04532327351522358604131570139, 7.74398608326178337806970095846, 8.955643677804340639523778996300

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