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Properties

Label 2-3248-203.6-c0-0-0
Degree $2$
Conductor $3248$
Sign $0.357 - 0.934i$
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  + (−0.222 + 0.974i)7-s + (0.900 + 0.433i)9-s + (0.376 + 0.781i)11-s + (−0.277 − 0.347i)23-s + (−0.222 − 0.974i)25-s + (0.222 + 0.974i)29-s + (−0.678 + 1.40i)37-s + (−0.900 − 0.433i)49-s + (−1.12 + 1.40i)53-s + (−0.623 + 0.781i)63-s + (0.400 + 0.193i)67-s + (1.12 − 0.541i)71-s + (−0.846 + 0.193i)77-s + (0.376 − 0.781i)79-s + (0.623 + 0.781i)81-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)7-s + (0.900 + 0.433i)9-s + (0.376 + 0.781i)11-s + (−0.277 − 0.347i)23-s + (−0.222 − 0.974i)25-s + (0.222 + 0.974i)29-s + (−0.678 + 1.40i)37-s + (−0.900 − 0.433i)49-s + (−1.12 + 1.40i)53-s + (−0.623 + 0.781i)63-s + (0.400 + 0.193i)67-s + (1.12 − 0.541i)71-s + (−0.846 + 0.193i)77-s + (0.376 − 0.781i)79-s + (0.623 + 0.781i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3248\)    =    \(2^{4} \cdot 7 \cdot 29\)
Sign: $0.357 - 0.934i$
Analytic conductor: \(1.62096\)
Root analytic conductor: \(1.27317\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3248} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3248,\ (\ :0),\ 0.357 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.264255568\)
\(L(\frac12)\) \(\approx\) \(1.264255568\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.222 - 0.974i)T \)
29 \( 1 + (-0.222 - 0.974i)T \)
good3 \( 1 + (-0.900 - 0.433i)T^{2} \)
5 \( 1 + (0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.376 - 0.781i)T + (-0.623 + 0.781i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.900 + 0.433i)T^{2} \)
23 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (-0.222 - 0.974i)T^{2} \)
37 \( 1 + (0.678 - 1.40i)T + (-0.623 - 0.781i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.623 - 0.781i)T^{2} \)
53 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (-0.900 - 0.433i)T^{2} \)
67 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
71 \( 1 + (-1.12 + 0.541i)T + (0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.222 + 0.974i)T^{2} \)
79 \( 1 + (-0.376 + 0.781i)T + (-0.623 - 0.781i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.222 - 0.974i)T^{2} \)
97 \( 1 + (-0.900 + 0.433i)T^{2} \)
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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

−8.958535430600292714106593189108, −8.282217675240105798364042325683, −7.47363590818375023161561124943, −6.68290604469608675209232229196, −6.09369047018674948919810364532, −4.98122036037618858423559278721, −4.52732743907697331027155568157, −3.41692099026906464755656954287, −2.37750061089451862226181574377, −1.54285825530847634013732895610, 0.808949316077380735617971902848, 1.92882671218186995726524193540, 3.44338176199247014893268001369, 3.81171059014286633142959704676, 4.73662916046925183358453560516, 5.76038718945470313574510302159, 6.53982792378210888558529631643, 7.18520896450438175698033609340, 7.83292294740137829861480640739, 8.699617340441510275370001436011

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