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Properties

Label 2-352-11.3-c1-0-4
Degree $2$
Conductor $352$
Sign $0.878 + 0.477i$
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  + (−0.212 − 0.653i)3-s + (−0.607 − 0.441i)5-s + (−0.944 + 2.90i)7-s + (2.04 − 1.48i)9-s + (3.27 − 0.510i)11-s + (5.02 − 3.64i)13-s + (−0.159 + 0.490i)15-s + (2.49 + 1.81i)17-s + (−1.30 − 4.02i)19-s + 2.09·21-s + 0.264·23-s + (−1.37 − 4.21i)25-s + (−3.07 − 2.23i)27-s + (−0.513 + 1.57i)29-s + (−3.39 + 2.46i)31-s + ⋯
L(s)  = 1  + (−0.122 − 0.377i)3-s + (−0.271 − 0.197i)5-s + (−0.356 + 1.09i)7-s + (0.681 − 0.495i)9-s + (0.988 − 0.153i)11-s + (1.39 − 1.01i)13-s + (−0.0411 + 0.126i)15-s + (0.604 + 0.439i)17-s + (−0.300 − 0.924i)19-s + 0.457·21-s + 0.0551·23-s + (−0.274 − 0.843i)25-s + (−0.591 − 0.429i)27-s + (−0.0952 + 0.293i)29-s + (−0.609 + 0.443i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 + 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.878 + 0.477i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.878 + 0.477i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30638 - 0.331976i\)
\(L(\frac12)\) \(\approx\) \(1.30638 - 0.331976i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.27 + 0.510i)T \)
good3 \( 1 + (0.212 + 0.653i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (0.607 + 0.441i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.944 - 2.90i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-5.02 + 3.64i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.49 - 1.81i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.30 + 4.02i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 0.264T + 23T^{2} \)
29 \( 1 + (0.513 - 1.57i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.39 - 2.46i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.730 - 2.24i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.75 - 5.40i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 9.90T + 43T^{2} \)
47 \( 1 + (-3.56 - 10.9i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.56 - 2.59i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.258 + 0.795i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.06 + 3.68i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.76T + 67T^{2} \)
71 \( 1 + (8.38 + 6.08i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (2.61 - 8.05i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.9 + 8.65i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (8.89 + 6.46i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 15.9T + 89T^{2} \)
97 \( 1 + (-0.712 + 0.518i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.53564141666253845845202093800, −10.58614212346309099754543108698, −9.334444958011983244020150456066, −8.726414480271267955617050968484, −7.66625406446278572715803017969, −6.36054543764097806363802591218, −5.86618623738369048595353038582, −4.25672846640901186254303435613, −3.08086300463060817169677127945, −1.22652799447617814410504899633, 1.49727122205353816728122679298, 3.86932828898630329915412898891, 4.03316463536418855775401602707, 5.74972688702494134816937275467, 6.91329608172213728943230432962, 7.56425139784204039372077840409, 8.932863132896721281724733392574, 9.792629330716071804561919765323, 10.68247512711097345909767562619, 11.34085069178250904962233246928

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