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Properties

Label 2-352-88.43-c1-0-1
Degree $2$
Conductor $352$
Sign $0.426 - 0.904i$
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  − 2·3-s + 9-s + (3 + 1.41i)11-s + 5.65i·17-s + 8.48i·19-s + 5·25-s + 4·27-s + (−6 − 2.82i)33-s + 11.3i·41-s − 8.48i·43-s − 7·49-s − 11.3i·51-s − 16.9i·57-s + 6·59-s − 14·67-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.333·9-s + (0.904 + 0.426i)11-s + 1.37i·17-s + 1.94i·19-s + 25-s + 0.769·27-s + (−1.04 − 0.492i)33-s + 1.76i·41-s − 1.29i·43-s − 49-s − 1.58i·51-s − 2.24i·57-s + 0.781·59-s − 1.71·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.426 - 0.904i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.426 - 0.904i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694545 + 0.440437i\)
\(L(\frac12)\) \(\approx\) \(0.694545 + 0.440437i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3 - 1.41i)T \)
good3 \( 1 + 2T + 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 5.65iT - 17T^{2} \)
19 \( 1 - 8.48iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 + 8.48iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 6T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 14T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 16.9iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 2.82iT - 83T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
97 \( 1 - 10T + 97T^{2} \)
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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

−11.78564837200667383413496578252, −10.69630340124542091396161148176, −10.12212387802391454264212625168, −8.878008511181446347730900255814, −7.83628813223193838458185988316, −6.50066287610632994931038182472, −5.98742920335379963792988417187, −4.79528488318644548625366106906, −3.62988165438279953775200850743, −1.51828845451592432983997148942, 0.71747212566875794729493289639, 2.90341150757303136225476350856, 4.56185882949164792426677026649, 5.37579875239465540589284238388, 6.52660976618871657974873541775, 7.14693182699885038244888899715, 8.715914008706220274171438711728, 9.437006067333637949281575212086, 10.71754742669555316812319664663, 11.38335954939917094632035276231

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