Properties

Label 2-8624-1.1-c1-0-83
Degree $2$
Conductor $8624$
Sign $1$
Analytic cond. $68.8629$
Root an. cond. $8.29837$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.41·3-s − 0.585·5-s + 2.82·9-s − 11-s − 3.82·13-s − 1.41·15-s + 3.65·17-s + 0.585·19-s + 6.24·23-s − 4.65·25-s − 0.414·27-s + 2.65·29-s + 4·31-s − 2.41·33-s − 9.41·37-s − 9.24·39-s − 5.41·41-s + 5.65·43-s − 1.65·45-s + 10.4·47-s + 8.82·51-s + 7.89·53-s + 0.585·55-s + 1.41·57-s + 5.58·59-s + 11.8·61-s + 2.24·65-s + ⋯
L(s)  = 1  + 1.39·3-s − 0.261·5-s + 0.942·9-s − 0.301·11-s − 1.06·13-s − 0.365·15-s + 0.886·17-s + 0.134·19-s + 1.30·23-s − 0.931·25-s − 0.0797·27-s + 0.493·29-s + 0.718·31-s − 0.420·33-s − 1.54·37-s − 1.48·39-s − 0.845·41-s + 0.862·43-s − 0.246·45-s + 1.52·47-s + 1.23·51-s + 1.08·53-s + 0.0789·55-s + 0.187·57-s + 0.727·59-s + 1.51·61-s + 0.278·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8624\)    =    \(2^{4} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(68.8629\)
Root analytic conductor: \(8.29837\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{8624} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8624,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.167987527\)
\(L(\frac12)\) \(\approx\) \(3.167987527\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + T \)
good3 \( 1 - 2.41T + 3T^{2} \)
5 \( 1 + 0.585T + 5T^{2} \)
13 \( 1 + 3.82T + 13T^{2} \)
17 \( 1 - 3.65T + 17T^{2} \)
19 \( 1 - 0.585T + 19T^{2} \)
23 \( 1 - 6.24T + 23T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 9.41T + 37T^{2} \)
41 \( 1 + 5.41T + 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 - 7.89T + 53T^{2} \)
59 \( 1 - 5.58T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 + 2.75T + 67T^{2} \)
71 \( 1 - 11.0T + 71T^{2} \)
73 \( 1 + 9.41T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 - 12.1T + 83T^{2} \)
89 \( 1 - 12.4T + 89T^{2} \)
97 \( 1 + 3.82T + 97T^{2} \)
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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

−7.74607036958580384323843468894, −7.35909382414697767814775652248, −6.67644969272791938895610773565, −5.50903971801855883138143774186, −4.99858711854964416335235212352, −3.99661333331763794510126724146, −3.40177385325507443005749039330, −2.65649552518627498150037535953, −2.06826634758642928235302460192, −0.799094839644138814639646614783, 0.799094839644138814639646614783, 2.06826634758642928235302460192, 2.65649552518627498150037535953, 3.40177385325507443005749039330, 3.99661333331763794510126724146, 4.99858711854964416335235212352, 5.50903971801855883138143774186, 6.67644969272791938895610773565, 7.35909382414697767814775652248, 7.74607036958580384323843468894

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