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Properties

Label 2-1248-39.38-c0-0-3
Degree $2$
Conductor $1248$
Sign $i$
Analytic cond. $0.622833$
Root an. cond. $0.789197$
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  − i·3-s + 1.41·5-s − 1.41i·7-s − 9-s − 13-s − 1.41i·15-s − 1.41i·19-s − 1.41·21-s + 2i·23-s + 1.00·25-s + i·27-s + 1.41i·31-s − 2.00i·35-s + i·39-s + 1.41·41-s + ⋯
L(s)  = 1  − i·3-s + 1.41·5-s − 1.41i·7-s − 9-s − 13-s − 1.41i·15-s − 1.41i·19-s − 1.41·21-s + 2i·23-s + 1.00·25-s + i·27-s + 1.41i·31-s − 2.00i·35-s + i·39-s + 1.41·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $i$
Analytic conductor: \(0.622833\)
Root analytic conductor: \(0.789197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.237933488\)
\(L(\frac12)\) \(\approx\) \(1.237933488\)
\(L(1)\) 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 \( 1 + iT \)
13 \( 1 + T \)
good5 \( 1 - 1.41T + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.41iT - T^{2} \)
23 \( 1 - 2iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 1.41iT - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.593477993656940468115305850093, −9.080983891341659409637587681372, −7.75420656098609521500049161507, −7.19034173378948048609927652356, −6.56245862159902089965630904239, −5.57236404137647117963665766618, −4.78767446435944540161859955083, −3.27697713813077974469272519270, −2.19380044167029038823823344074, −1.14526597085907940660257118945, 2.22054860572870452195712263468, 2.67823078617822368121724589253, 4.20747543092371571245110635950, 5.18943903423088418241553756889, 5.83140670259967416911420751374, 6.34459563294790264528625793024, 7.913668964434451360218617702953, 8.793578220797358060691257982315, 9.387486045480286963215546183805, 9.997803497634161503938452659034

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