↑

Properties

Label 2-352-44.43-c1-0-6
Degree $2$
Conductor $352$
Sign $0.834 + 0.550i$
Analytic cond. $2.81073$
Root an. cond. $1.67652$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 0.529i·3-s − 2.77·5-s + 3.18·7-s + 2.71·9-s + (0.665 + 3.24i)11-s − 6.00i·13-s + 1.47i·15-s − 4.67i·17-s + 7.50·19-s − 1.68i·21-s − 5.02i·23-s + 2.71·25-s − 3.02i·27-s + 5.03i·29-s + 3.96i·31-s + ⋯
L(s)  = 1  − 0.305i·3-s − 1.24·5-s + 1.20·7-s + 0.906·9-s + (0.200 + 0.979i)11-s − 1.66i·13-s + 0.379i·15-s − 1.13i·17-s + 1.72·19-s − 0.367i·21-s − 1.04i·23-s + 0.543·25-s − 0.582i·27-s + 0.934i·29-s + 0.712i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.834 + 0.550i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.834 + 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26774 - 0.380607i\)
\(L(\frac12)\) \(\approx\) \(1.26774 - 0.380607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (-0.665 - 3.24i)T \)
good3 \( 1 + 0.529iT - 3T^{2} \)
5 \( 1 + 2.77T + 5T^{2} \)
7 \( 1 - 3.18T + 7T^{2} \)
13 \( 1 + 6.00iT - 13T^{2} \)
17 \( 1 + 4.67iT - 17T^{2} \)
19 \( 1 - 7.50T + 19T^{2} \)
23 \( 1 + 5.02iT - 23T^{2} \)
29 \( 1 - 5.03iT - 29T^{2} \)
31 \( 1 - 3.96iT - 31T^{2} \)
37 \( 1 + 2.77T + 37T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 - 1.33T + 43T^{2} \)
47 \( 1 - 8.61iT - 47T^{2} \)
53 \( 1 - 0.117T + 53T^{2} \)
59 \( 1 + 6.41iT - 59T^{2} \)
61 \( 1 + 4.32iT - 61T^{2} \)
67 \( 1 - 10.5iT - 67T^{2} \)
71 \( 1 + 1.91iT - 71T^{2} \)
73 \( 1 + 7.34iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 7.69T + 83T^{2} \)
89 \( 1 - 1.95T + 89T^{2} \)
97 \( 1 + 13.3T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.53078509796735867349501690927, −10.59926790147257269889554721442, −9.612600515107934967421282416071, −8.229029871724027992970254944607, −7.61177114307544771086676721557, −7.04474804935171409949524980612, −5.17189428594427658824229873319, −4.50110730566927097895969893616, −3.07471915989349300059710040789, −1.16751638528129477921871070377, 1.53575223585370216916168653561, 3.74233511904721328525009843824, 4.27225032014185792324947059741, 5.53893932507849803931166704769, 7.06398011960465920630869206297, 7.79415019637989975101379066572, 8.671129086187156632144687214532, 9.682541172027293157331795376934, 10.92221180764582702954265060676, 11.62124500138722377379424774724

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