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Properties

Label 2-352-88.37-c1-0-1
Degree $2$
Conductor $352$
Sign $0.159 - 0.987i$
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  + (−0.826 − 0.268i)3-s + (2.15 + 2.97i)5-s + (0.369 + 1.13i)7-s + (−1.81 − 1.31i)9-s + (−2.03 + 2.62i)11-s + (−2.33 + 3.22i)13-s + (−0.986 − 3.03i)15-s + (2.57 − 1.86i)17-s + (4.09 + 1.33i)19-s − 1.03i·21-s − 2.46·23-s + (−2.62 + 8.08i)25-s + (2.67 + 3.68i)27-s + (−3.65 + 1.18i)29-s + (7.40 + 5.37i)31-s + ⋯
L(s)  = 1  + (−0.476 − 0.154i)3-s + (0.965 + 1.32i)5-s + (0.139 + 0.429i)7-s + (−0.605 − 0.439i)9-s + (−0.612 + 0.790i)11-s + (−0.648 + 0.893i)13-s + (−0.254 − 0.783i)15-s + (0.623 − 0.453i)17-s + (0.939 + 0.305i)19-s − 0.226i·21-s − 0.513·23-s + (−0.525 + 1.61i)25-s + (0.515 + 0.709i)27-s + (−0.678 + 0.220i)29-s + (1.32 + 0.965i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.159 - 0.987i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.159 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886780 + 0.754899i\)
\(L(\frac12)\) \(\approx\) \(0.886780 + 0.754899i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 - 2.62i)T \)
good3 \( 1 + (0.826 + 0.268i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-2.15 - 2.97i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.369 - 1.13i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.33 - 3.22i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.57 + 1.86i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-4.09 - 1.33i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 2.46T + 23T^{2} \)
29 \( 1 + (3.65 - 1.18i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.40 - 5.37i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-5.06 + 1.64i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.164 + 0.506i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.51iT - 43T^{2} \)
47 \( 1 + (0.110 - 0.341i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.42 - 1.96i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.566 + 0.184i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.28 + 5.89i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 11.2iT - 67T^{2} \)
71 \( 1 + (-4.52 + 3.28i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.825 + 2.54i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (9.30 + 6.75i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.589 + 0.811i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + (-4.81 - 3.49i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.77668482850698527358798941386, −10.71233951754887431392562277033, −9.910748952801114186415191941780, −9.210079646457421144460554831018, −7.64051335105587768244007967263, −6.80174137513423330687492737621, −5.93388084092272626962903803192, −5.04807496960307422197684502372, −3.15469731509560581993470434747, −2.10984562787973340211577485218, 0.867215513219699449907121849641, 2.70134427583386621394985670301, 4.54849520290496451742288363042, 5.51440889773300685885639279256, 5.89619034374636970156100213056, 7.77911507691093825161251959872, 8.373947843932854325231776451208, 9.620156892064580325935829418368, 10.22175093346658483844458698267, 11.27280711541998825507960024852

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