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Properties

Label 2-352-44.43-c1-0-11
Degree $2$
Conductor $352$
Sign $-0.995 + 0.0985i$
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  − 2.81i·3-s + 0.289·5-s − 4.38·7-s − 4.91·9-s + (−2.56 − 2.10i)11-s + 1.55i·13-s − 0.813i·15-s − 3.57i·17-s + 6.39·19-s + 12.3i·21-s + 3.39i·23-s − 4.91·25-s + 5.39i·27-s − 3.64i·29-s − 9.01i·31-s + ⋯
L(s)  = 1  − 1.62i·3-s + 0.129·5-s − 1.65·7-s − 1.63·9-s + (−0.773 − 0.634i)11-s + 0.432i·13-s − 0.210i·15-s − 0.865i·17-s + 1.46·19-s + 2.69i·21-s + 0.707i·23-s − 0.983·25-s + 1.03i·27-s − 0.677i·29-s − 1.61i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $-0.995 + 0.0985i$
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: \((2,\ 352,\ (\ :1/2),\ -0.995 + 0.0985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0372802 - 0.755043i\)
\(L(\frac12)\) \(\approx\) \(0.0372802 - 0.755043i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.56 + 2.10i)T \)
good3 \( 1 + 2.81iT - 3T^{2} \)
5 \( 1 - 0.289T + 5T^{2} \)
7 \( 1 + 4.38T + 7T^{2} \)
13 \( 1 - 1.55iT - 13T^{2} \)
17 \( 1 + 3.57iT - 17T^{2} \)
19 \( 1 - 6.39T + 19T^{2} \)
23 \( 1 - 3.39iT - 23T^{2} \)
29 \( 1 + 3.64iT - 29T^{2} \)
31 \( 1 + 9.01iT - 31T^{2} \)
37 \( 1 - 0.289T + 37T^{2} \)
41 \( 1 + 6.68iT - 41T^{2} \)
43 \( 1 + 5.12T + 43T^{2} \)
47 \( 1 - 7.04iT - 47T^{2} \)
53 \( 1 - 9.25T + 53T^{2} \)
59 \( 1 - 0.440iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + 3.97iT - 67T^{2} \)
71 \( 1 + 5.76iT - 71T^{2} \)
73 \( 1 - 6.68iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 - 13.9T + 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 + 8.75T + 97T^{2} \)
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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

−11.42393404986822566471655268493, −9.907544355789018836664141361864, −9.236001443810685077099698972718, −7.85200045020418312565015444659, −7.25654436004095670790524323100, −6.26607024715996101673746194777, −5.58597501368552018587045842982, −3.41602910432500097165539468459, −2.36684652880227041738922267683, −0.49560360651987670943242247853, 2.94783225097990509282968483657, 3.70726584608862198874254727620, 4.98886521472958553150454558624, 5.85858554716848506782556412213, 7.09980924244429776580275927491, 8.526523702005673351586951687478, 9.471063307422630497483564959939, 10.20251806703942273271044280417, 10.41962644499434180879911176648, 11.85517212116258456359219016522

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