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Properties

Label 1-356-356.335-r1-0-0
Degree $1$
Conductor $356$
Sign $0.184 - 0.982i$
Analytic cond. $38.2575$
Root an. cond. $38.2575$
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  + (0.909 − 0.415i)3-s + (0.142 + 0.989i)5-s + (−0.989 + 0.142i)7-s + (0.654 − 0.755i)9-s + (0.142 − 0.989i)11-s + (0.909 − 0.415i)13-s + (0.540 + 0.841i)15-s + (−0.841 − 0.540i)17-s + (−0.755 − 0.654i)19-s + (−0.841 + 0.540i)21-s + (0.755 + 0.654i)23-s + (−0.959 + 0.281i)25-s + (0.281 − 0.959i)27-s + (−0.989 + 0.142i)29-s + (0.755 − 0.654i)31-s + ⋯
L(s)  = 1  + (0.909 − 0.415i)3-s + (0.142 + 0.989i)5-s + (−0.989 + 0.142i)7-s + (0.654 − 0.755i)9-s + (0.142 − 0.989i)11-s + (0.909 − 0.415i)13-s + (0.540 + 0.841i)15-s + (−0.841 − 0.540i)17-s + (−0.755 − 0.654i)19-s + (−0.841 + 0.540i)21-s + (0.755 + 0.654i)23-s + (−0.959 + 0.281i)25-s + (0.281 − 0.959i)27-s + (−0.989 + 0.142i)29-s + (0.755 − 0.654i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(356\)    =    \(2^{2} \cdot 89\)
Sign: $0.184 - 0.982i$
Analytic conductor: \(38.2575\)
Root analytic conductor: \(38.2575\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{356} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 356,\ (1:\ ),\ 0.184 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.683480442 - 1.397325087i\)
\(L(\frac12)\) \(\approx\) \(1.683480442 - 1.397325087i\)
\(L(1)\) \(\approx\) \(1.313178780 - 0.2719061593i\)
\(L(1)\) \(\approx\) \(1.313178780 - 0.2719061593i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
89 \( 1 \)
good3 \( 1 + (0.909 - 0.415i)T \)
5 \( 1 + (0.142 + 0.989i)T \)
7 \( 1 + (-0.989 + 0.142i)T \)
11 \( 1 + (0.142 - 0.989i)T \)
13 \( 1 + (0.909 - 0.415i)T \)
17 \( 1 + (-0.841 - 0.540i)T \)
19 \( 1 + (-0.755 - 0.654i)T \)
23 \( 1 + (0.755 + 0.654i)T \)
29 \( 1 + (-0.989 + 0.142i)T \)
31 \( 1 + (0.755 - 0.654i)T \)
37 \( 1 - iT \)
41 \( 1 + (0.909 + 0.415i)T \)
43 \( 1 + (0.989 + 0.142i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (0.909 + 0.415i)T \)
61 \( 1 + (0.281 - 0.959i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
71 \( 1 + (-0.142 + 0.989i)T \)
73 \( 1 + (-0.654 - 0.755i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (0.540 - 0.841i)T \)
97 \( 1 + (-0.142 - 0.989i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.996094001418556903904996361732, −23.98482888405298458333818324946, −22.95382531066151240550051456792, −22.0384855515924317952628200751, −20.81611292383011316413805571249, −20.589070993732777071722196763477, −19.50636048800102072248206864267, −18.91329405482463275805271577146, −17.46613929878838434616397059721, −16.55432027023837642721813323732, −15.80449418413306431925541891286, −15.00524479491042682099560461278, −13.83291108774489836402935106694, −12.99202384987231859037493156554, −12.46321163865417963553382771104, −10.81938289896410801158595768093, −9.84926850633020395560055212925, −9.0486471962885323623482059978, −8.41412484739300258099325537141, −7.10812967395333017334192260320, −5.99658332730532401914154168567, −4.46911754161623015357222600690, −3.96234113272845305468094638820, −2.514943510614118913799577607592, −1.35536185260963520939404213225, 0.55432498560833873779863744290, 2.27877719640444499680887570641, 3.11588151340811154066478183609, 3.88650255814669710768469012559, 5.9076986636499926594454478967, 6.62412101821017961521474452297, 7.52685747610851320767661125281, 8.77974667840477368262963796491, 9.420793611114785804501737149426, 10.66379362875034091065691926066, 11.489742817865852386803650485524, 13.1250825275990810645390814061, 13.32831322318143595784849823701, 14.41239674722982790444513920972, 15.37100654173888674934688393655, 16.00465293771162495578103474419, 17.47786359360469008913526991522, 18.421844359155654303286559050138, 19.12178159884084094913915954808, 19.646961008380431912119763642869, 20.842738476542012797140694570657, 21.71575221036305571597032770818, 22.608455080082958543505024965270, 23.44643984320503699879932656006, 24.569961451779250923029935629291

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