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Properties

Label 2-352-88.83-c1-0-6
Degree $2$
Conductor $352$
Sign $0.976 - 0.215i$
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  + (1.63 + 1.18i)3-s + (−1.62 − 0.527i)5-s + (3.70 − 2.68i)7-s + (0.333 + 1.02i)9-s + (2.77 + 1.81i)11-s + (−0.147 − 0.455i)13-s + (−2.02 − 2.79i)15-s + (2.14 + 0.696i)17-s + (−1.24 + 1.70i)19-s + 9.24·21-s + 7.45i·23-s + (−1.68 − 1.22i)25-s + (1.19 − 3.68i)27-s + (−3.30 + 2.40i)29-s + (−3.84 + 1.24i)31-s + ⋯
L(s)  = 1  + (0.943 + 0.685i)3-s + (−0.726 − 0.236i)5-s + (1.39 − 1.01i)7-s + (0.111 + 0.342i)9-s + (0.836 + 0.547i)11-s + (−0.0410 − 0.126i)13-s + (−0.523 − 0.720i)15-s + (0.519 + 0.168i)17-s + (−0.284 + 0.391i)19-s + 2.01·21-s + 1.55i·23-s + (−0.337 − 0.244i)25-s + (0.230 − 0.709i)27-s + (−0.614 + 0.446i)29-s + (−0.690 + 0.224i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.976 - 0.215i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.976 - 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82442 + 0.198929i\)
\(L(\frac12)\) \(\approx\) \(1.82442 + 0.198929i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.77 - 1.81i)T \)
good3 \( 1 + (-1.63 - 1.18i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (1.62 + 0.527i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-3.70 + 2.68i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (0.147 + 0.455i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.14 - 0.696i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.24 - 1.70i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 - 7.45iT - 23T^{2} \)
29 \( 1 + (3.30 - 2.40i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.84 - 1.24i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.89 + 5.36i)T + (-11.4 + 35.1i)T^{2} \)
41 \( 1 + (3.72 - 5.12i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.32iT - 43T^{2} \)
47 \( 1 + (-2.59 + 3.56i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.67 + 0.545i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-6.30 + 4.58i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (0.782 - 2.40i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 4.41T + 67T^{2} \)
71 \( 1 + (4.63 + 1.50i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.11 + 5.65i)T + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (-5.15 - 15.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (5.14 + 1.67i)T + (67.1 + 48.7i)T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 + (-1.68 - 5.17i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.48687914507304626772732695050, −10.52420882743968963736655367738, −9.610851740068528235475136925062, −8.645600257669039453801862241284, −7.86687989928302382979864525869, −7.15323444985431647796746314407, −5.29333731524742810817559270573, −4.07508808628458179286058896073, −3.71040767540842620908771391603, −1.62680174736185235223632349853, 1.72130784877759638776693313205, 2.88587446305526921560828621295, 4.27726343701190746222562498586, 5.57860083968228509986276774725, 6.94064304227840441442972793755, 7.910515292097411973280577716157, 8.480759140006065559365978087151, 9.150682636038027336167383531497, 10.78565182321918911897729166968, 11.61212575290281280667854334063

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