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Properties

Label 2-1376-344.195-c0-0-0
Degree $2$
Conductor $1376$
Sign $0.931 + 0.363i$
Analytic cond. $0.686713$
Root an. cond. $0.828681$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.162 + 0.414i)3-s + (0.587 − 0.545i)9-s + (−0.440 − 1.92i)11-s + (−0.109 − 1.46i)17-s + (1.40 + 1.29i)19-s + (−0.988 − 0.149i)25-s + (0.722 + 0.347i)27-s + (0.727 − 0.495i)33-s + (1.03 + 1.29i)41-s + (−0.365 + 0.930i)43-s + (−0.5 + 0.866i)49-s + (0.587 − 0.283i)51-s + (−0.310 + 0.791i)57-s + (0.658 + 0.317i)59-s + (−0.733 − 0.680i)67-s + ⋯
L(s)  = 1  + (0.162 + 0.414i)3-s + (0.587 − 0.545i)9-s + (−0.440 − 1.92i)11-s + (−0.109 − 1.46i)17-s + (1.40 + 1.29i)19-s + (−0.988 − 0.149i)25-s + (0.722 + 0.347i)27-s + (0.727 − 0.495i)33-s + (1.03 + 1.29i)41-s + (−0.365 + 0.930i)43-s + (−0.5 + 0.866i)49-s + (0.587 − 0.283i)51-s + (−0.310 + 0.791i)57-s + (0.658 + 0.317i)59-s + (−0.733 − 0.680i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1376\)    =    \(2^{5} \cdot 43\)
Sign: $0.931 + 0.363i$
Analytic conductor: \(0.686713\)
Root analytic conductor: \(0.828681\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1376} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1376,\ (\ :0),\ 0.931 + 0.363i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.172609427\)
\(L(\frac12)\) \(\approx\) \(1.172609427\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 + (0.365 - 0.930i)T \)
good3 \( 1 + (-0.162 - 0.414i)T + (-0.733 + 0.680i)T^{2} \)
5 \( 1 + (0.988 + 0.149i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.440 + 1.92i)T + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (-0.365 - 0.930i)T^{2} \)
17 \( 1 + (0.109 + 1.46i)T + (-0.988 + 0.149i)T^{2} \)
19 \( 1 + (-1.40 - 1.29i)T + (0.0747 + 0.997i)T^{2} \)
23 \( 1 + (-0.826 - 0.563i)T^{2} \)
29 \( 1 + (0.733 + 0.680i)T^{2} \)
31 \( 1 + (-0.955 + 0.294i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-1.03 - 1.29i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.658 - 0.317i)T + (0.623 + 0.781i)T^{2} \)
61 \( 1 + (-0.955 - 0.294i)T^{2} \)
67 \( 1 + (0.733 + 0.680i)T + (0.0747 + 0.997i)T^{2} \)
71 \( 1 + (-0.826 + 0.563i)T^{2} \)
73 \( 1 + (0.826 + 0.563i)T + (0.365 + 0.930i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.455 + 1.16i)T + (-0.733 + 0.680i)T^{2} \)
89 \( 1 + (-0.603 - 1.53i)T + (-0.733 + 0.680i)T^{2} \)
97 \( 1 + (0.162 + 0.712i)T + (-0.900 + 0.433i)T^{2} \)
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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

−9.656973841267569333380322499316, −9.104758470696042337402293641178, −8.063616760004862354660454202133, −7.50104751517680674764516472286, −6.27692845194144639580758329375, −5.64381172696253198063873611018, −4.62774874494108411479795231234, −3.50519903291258897413177294888, −2.93636174871354404973200380918, −1.09150959803001508814673030050, 1.66354291776406214765805938279, 2.43739426212892491727553573317, 3.91093588903170774340929122812, 4.75857133546542325252606191899, 5.59428298060141500042050181033, 6.90990185876952542863947301394, 7.31551814358399727482897881802, 8.028372892364556229446862658413, 9.059952153905686198891867110794, 9.949461149240437671436828370836

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