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Properties

Label 2-352-44.43-c1-0-8
Degree $2$
Conductor $352$
Sign $0.737 + 0.675i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.34i·3-s + 2.48·5-s + 1.62·7-s + 1.19·9-s + (−3.31 + 0.146i)11-s − 1.20i·13-s − 3.34i·15-s + 5.41i·17-s + 2.59·19-s − 2.17i·21-s − 3.63i·23-s + 1.19·25-s − 5.63i·27-s − 9.86i·29-s + 0.949i·31-s + ⋯
L(s)  = 1  − 0.775i·3-s + 1.11·5-s + 0.612·7-s + 0.398·9-s + (−0.999 + 0.0441i)11-s − 0.334i·13-s − 0.863i·15-s + 1.31i·17-s + 0.594·19-s − 0.475i·21-s − 0.758i·23-s + 0.239·25-s − 1.08i·27-s − 1.83i·29-s + 0.170i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.737 + 0.675i$
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: \((2,\ 352,\ (\ :1/2),\ 0.737 + 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55411 - 0.603906i\)
\(L(\frac12)\) \(\approx\) \(1.55411 - 0.603906i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.31 - 0.146i)T \)
good3 \( 1 + 1.34iT - 3T^{2} \)
5 \( 1 - 2.48T + 5T^{2} \)
7 \( 1 - 1.62T + 7T^{2} \)
13 \( 1 + 1.20iT - 13T^{2} \)
17 \( 1 - 5.41iT - 17T^{2} \)
19 \( 1 - 2.59T + 19T^{2} \)
23 \( 1 + 3.63iT - 23T^{2} \)
29 \( 1 + 9.86iT - 29T^{2} \)
31 \( 1 - 0.949iT - 31T^{2} \)
37 \( 1 - 2.48T + 37T^{2} \)
41 \( 1 - 7.83iT - 41T^{2} \)
43 \( 1 + 6.62T + 43T^{2} \)
47 \( 1 - 5.66iT - 47T^{2} \)
53 \( 1 + 7.37T + 53T^{2} \)
59 \( 1 - 12.0iT - 59T^{2} \)
61 \( 1 - 0.969iT - 61T^{2} \)
67 \( 1 - 8.61iT - 67T^{2} \)
71 \( 1 - 14.3iT - 71T^{2} \)
73 \( 1 + 7.83iT - 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 3.38T + 83T^{2} \)
89 \( 1 + 14.5T + 89T^{2} \)
97 \( 1 - 6.15T + 97T^{2} \)
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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

−11.38038449401192368503681153247, −10.29501073263614205884511526823, −9.763268314934401104632402802494, −8.283059176008742635942225093794, −7.73024417056810850673629659054, −6.44467527260490501994568522880, −5.69365091250351660215785964238, −4.47402137573043558438952907493, −2.56180405490412421782998039286, −1.47946574280253193631971354326, 1.82470409520826490759343552209, 3.32012128631068148055953057266, 4.97707457851921448343928598980, 5.28024303877725487708874790501, 6.81101707608184001856871100029, 7.82661256359416002486453743760, 9.183502332390903969699393992776, 9.696873604636664569458261620402, 10.56097459535211848947169064408, 11.33093501004551857450647624464

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