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Properties

Label 2-3248-3248.1371-c0-0-1
Degree $2$
Conductor $3248$
Sign $-0.724 + 0.689i$
Analytic cond. $1.62096$
Root an. cond. $1.27317$
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  + (−0.974 − 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.623 − 0.781i)7-s + (−0.781 − 0.623i)8-s + (0.222 − 0.974i)9-s + (0.0990 + 0.433i)11-s + (0.433 + 0.900i)14-s + (0.623 + 0.781i)16-s + (−0.433 + 0.900i)18-s − 0.445i·22-s + (−1.40 − 0.678i)23-s + (−0.781 − 0.623i)25-s + (−0.222 − 0.974i)28-s + (0.781 + 0.623i)29-s + (−0.433 − 0.900i)32-s + ⋯
L(s)  = 1  + (−0.974 − 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.623 − 0.781i)7-s + (−0.781 − 0.623i)8-s + (0.222 − 0.974i)9-s + (0.0990 + 0.433i)11-s + (0.433 + 0.900i)14-s + (0.623 + 0.781i)16-s + (−0.433 + 0.900i)18-s − 0.445i·22-s + (−1.40 − 0.678i)23-s + (−0.781 − 0.623i)25-s + (−0.222 − 0.974i)28-s + (0.781 + 0.623i)29-s + (−0.433 − 0.900i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.724 + 0.689i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3248\)    =    \(2^{4} \cdot 7 \cdot 29\)
Sign: $-0.724 + 0.689i$
Analytic conductor: \(1.62096\)
Root analytic conductor: \(1.27317\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3248} (1371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3248,\ (\ :0),\ -0.724 + 0.689i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4857302271\)
\(L(\frac12)\) \(\approx\) \(0.4857302271\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.974 + 0.222i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (-0.781 - 0.623i)T \)
good3 \( 1 + (-0.222 + 0.974i)T^{2} \)
5 \( 1 + (0.781 + 0.623i)T^{2} \)
11 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.433 + 0.900i)T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + (-0.222 - 0.974i)T^{2} \)
23 \( 1 + (1.40 + 0.678i)T + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.781 + 0.623i)T^{2} \)
37 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (0.433 + 0.900i)T^{2} \)
53 \( 1 + (1.00 + 0.351i)T + (0.781 + 0.623i)T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.222 - 0.974i)T^{2} \)
67 \( 1 + (0.119 + 0.189i)T + (-0.433 + 0.900i)T^{2} \)
71 \( 1 + (1.75 - 0.400i)T + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.781 - 0.623i)T^{2} \)
79 \( 1 + (0.566 + 0.900i)T + (-0.433 + 0.900i)T^{2} \)
83 \( 1 + (-0.974 + 0.222i)T^{2} \)
89 \( 1 + (0.781 + 0.623i)T^{2} \)
97 \( 1 + (0.974 - 0.222i)T^{2} \)
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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

−8.675137691603261588530466043675, −7.78990011369042505901492036814, −7.18780971124149122140150718675, −6.43607425107033928361535750204, −5.95871492927116247624757796466, −4.34467127100978390367589275376, −3.76019273938711597013411813092, −2.79751149489010250808362642237, −1.67013315026869209862449704218, −0.39269000606397556428996398437, 1.54520592885330843561989647062, 2.45312071749943158130974069066, 3.32825110508361389622980558004, 4.63499297496160172683160183051, 5.70773938991581841135505496232, 6.08125713895557979743950727664, 6.98558629915933085173009152538, 7.889874896906889848316399528747, 8.253559737825681082815250214471, 9.099371413813116497137421282410

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