Properties

Label 2-10304-1.1-c1-0-34
Degree $2$
Conductor $10304$
Sign $-1$
Analytic cond. $82.2778$
Root an. cond. $9.07071$
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  + 2·3-s + 7-s + 9-s + 4·11-s − 6·13-s + 2·21-s − 23-s − 5·25-s − 4·27-s − 2·29-s − 2·31-s + 8·33-s − 2·37-s − 12·39-s − 6·41-s + 4·43-s + 2·47-s + 49-s − 14·53-s + 14·59-s − 12·61-s + 63-s − 4·67-s − 2·69-s − 2·73-s − 10·75-s + 4·77-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 1.66·13-s + 0.436·21-s − 0.208·23-s − 25-s − 0.769·27-s − 0.371·29-s − 0.359·31-s + 1.39·33-s − 0.328·37-s − 1.92·39-s − 0.937·41-s + 0.609·43-s + 0.291·47-s + 1/7·49-s − 1.92·53-s + 1.82·59-s − 1.53·61-s + 0.125·63-s − 0.488·67-s − 0.240·69-s − 0.234·73-s − 1.15·75-s + 0.455·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10304\)    =    \(2^{6} \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(82.2778\)
Root analytic conductor: \(9.07071\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10304} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 10304,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 - T \)
23 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−16.93381721572475, −16.39496681831502, −15.37831921204087, −15.15214094802366, −14.40431967147312, −14.23189042194999, −13.68523964223713, −12.88722721723556, −12.24454300298356, −11.74115158949026, −11.18078464761274, −10.24161882083024, −9.614211531756728, −9.269773294776337, −8.620708591774556, −7.924350645152293, −7.455694229933261, −6.821331753807620, −5.947876138981294, −5.165935787005167, −4.355896444895042, −3.744875620122098, −2.963901416575064, −2.168873522599320, −1.552948187628666, 0, 1.552948187628666, 2.168873522599320, 2.963901416575064, 3.744875620122098, 4.355896444895042, 5.165935787005167, 5.947876138981294, 6.821331753807620, 7.455694229933261, 7.924350645152293, 8.620708591774556, 9.269773294776337, 9.614211531756728, 10.24161882083024, 11.18078464761274, 11.74115158949026, 12.24454300298356, 12.88722721723556, 13.68523964223713, 14.23189042194999, 14.40431967147312, 15.15214094802366, 15.37831921204087, 16.39496681831502, 16.93381721572475

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