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Properties

Label 2-352-88.69-c1-0-5
Degree $2$
Conductor $352$
Sign $-0.435 + 0.899i$
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  + (−2.62 + 0.852i)3-s + (−1.70 + 2.35i)5-s + (−0.730 + 2.24i)7-s + (3.72 − 2.70i)9-s + (2.02 − 2.62i)11-s + (−1.49 − 2.05i)13-s + (2.47 − 7.62i)15-s + (−4.77 − 3.46i)17-s + (−0.252 + 0.0821i)19-s − 6.51i·21-s − 4.18·23-s + (−1.06 − 3.27i)25-s + (−2.60 + 3.58i)27-s + (3.08 + 1.00i)29-s + (1.66 − 1.20i)31-s + ⋯
L(s)  = 1  + (−1.51 + 0.492i)3-s + (−0.763 + 1.05i)5-s + (−0.276 + 0.849i)7-s + (1.24 − 0.902i)9-s + (0.609 − 0.792i)11-s + (−0.414 − 0.571i)13-s + (0.639 − 1.96i)15-s + (−1.15 − 0.841i)17-s + (−0.0580 + 0.0188i)19-s − 1.42i·21-s − 0.871·23-s + (−0.212 − 0.654i)25-s + (−0.500 + 0.689i)27-s + (0.573 + 0.186i)29-s + (0.298 − 0.217i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 + 0.899i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.435 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $-0.435 + 0.899i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ -0.435 + 0.899i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0182238 - 0.0290778i\)
\(L(\frac12)\) \(\approx\) \(0.0182238 - 0.0290778i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 + 2.62i)T \)
good3 \( 1 + (2.62 - 0.852i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (1.70 - 2.35i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.730 - 2.24i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.49 + 2.05i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.77 + 3.46i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.252 - 0.0821i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.18T + 23T^{2} \)
29 \( 1 + (-3.08 - 1.00i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.66 + 1.20i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.740 + 0.240i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.927 - 2.85i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.14iT - 43T^{2} \)
47 \( 1 + (2.98 + 9.17i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.77 + 7.94i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (5.50 + 1.78i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.60 - 4.96i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.241iT - 67T^{2} \)
71 \( 1 + (3.19 + 2.31i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.96 - 6.04i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.19 - 3.04i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.95 - 12.3i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + 16.2T + 89T^{2} \)
97 \( 1 + (-5.90 + 4.29i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.35310354511687945725508504442, −10.53513237409731763849542498684, −9.615262618646243562757929998153, −8.382273543252043612144999112693, −6.98888420867791438143422187131, −6.30663364680373846291024601072, −5.38472385921767783366500435757, −4.19302399721496243499017760770, −2.89168236433642964152030015393, −0.02960509536396466725243145112, 1.46598680481628014020303114704, 4.28677744659701129389096928871, 4.59781759314430370506518067613, 6.10989686315967705246938138923, 6.87177124406594988712166964763, 7.78303771354705113312726191743, 9.006372384079482724992403816936, 10.16661782754662568559677365281, 11.06103515519904233233034709996, 11.93685934497144342153965255136

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