Properties

Label 1-344-344.163-r0-0-0
Degree $1$
Conductor $344$
Sign $0.512 + 0.858i$
Analytic cond. $1.59752$
Root an. cond. $1.59752$
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  + (−0.955 − 0.294i)3-s + (0.365 + 0.930i)5-s + (−0.5 − 0.866i)7-s + (0.826 + 0.563i)9-s + (−0.900 + 0.433i)11-s + (0.988 + 0.149i)13-s + (−0.0747 − 0.997i)15-s + (0.365 − 0.930i)17-s + (−0.826 + 0.563i)19-s + (0.222 + 0.974i)21-s + (−0.0747 + 0.997i)23-s + (−0.733 + 0.680i)25-s + (−0.623 − 0.781i)27-s + (0.955 − 0.294i)29-s + (0.733 + 0.680i)31-s + ⋯
L(s)  = 1  + (−0.955 − 0.294i)3-s + (0.365 + 0.930i)5-s + (−0.5 − 0.866i)7-s + (0.826 + 0.563i)9-s + (−0.900 + 0.433i)11-s + (0.988 + 0.149i)13-s + (−0.0747 − 0.997i)15-s + (0.365 − 0.930i)17-s + (−0.826 + 0.563i)19-s + (0.222 + 0.974i)21-s + (−0.0747 + 0.997i)23-s + (−0.733 + 0.680i)25-s + (−0.623 − 0.781i)27-s + (0.955 − 0.294i)29-s + (0.733 + 0.680i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.512 + 0.858i$
Analytic conductor: \(1.59752\)
Root analytic conductor: \(1.59752\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 344,\ (0:\ ),\ 0.512 + 0.858i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7052336293 + 0.4003375811i\)
\(L(\frac12)\) \(\approx\) \(0.7052336293 + 0.4003375811i\)
\(L(1)\) \(\approx\) \(0.7773287819 + 0.1126988197i\)
\(L(1)\) \(\approx\) \(0.7773287819 + 0.1126988197i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
43 \( 1 \)
good3 \( 1 + (-0.955 - 0.294i)T \)
5 \( 1 + (0.365 + 0.930i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (0.988 + 0.149i)T \)
17 \( 1 + (0.365 - 0.930i)T \)
19 \( 1 + (-0.826 + 0.563i)T \)
23 \( 1 + (-0.0747 + 0.997i)T \)
29 \( 1 + (0.955 - 0.294i)T \)
31 \( 1 + (0.733 + 0.680i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.222 + 0.974i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (0.988 - 0.149i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (-0.733 + 0.680i)T \)
67 \( 1 + (0.826 - 0.563i)T \)
71 \( 1 + (0.0747 + 0.997i)T \)
73 \( 1 + (0.988 + 0.149i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.955 + 0.294i)T \)
89 \( 1 + (-0.955 - 0.294i)T \)
97 \( 1 + (-0.900 + 0.433i)T \)
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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

−24.60404358762382308880316149625, −23.77741127673414416977258832516, −23.089331858328715467866801467245, −21.94238304806447917260315506013, −21.30932388452053516871226557450, −20.68080593154573941380774058500, −19.246972303460808851141968219250, −18.407362281671928321598849010203, −17.52549939059586495615715203259, −16.61850550745701394718741256663, −15.886668018737076652464279394189, −15.250712973071698925483649391635, −13.588021149168528387467493945154, −12.70129690209405921253602409701, −12.20482697560710187324965135393, −10.87768971792426458103892756738, −10.18106430686365796304771250912, −8.95312842848489777390592940597, −8.278591830331626143627912119485, −6.47109628416199575983585682389, −5.815378921252865018864002009207, −5.005936822393597092371443833109, −3.821348278644033314445952147218, −2.22765846476207192350680854391, −0.63956995402971003229358811126, 1.26276468069633575856678675594, 2.74760585960530424449364450784, 4.03422217587141357616750545714, 5.319623087909865383640296278516, 6.37481971772218036984256402688, 7.01425936489144667457361156309, 7.99562884881712480200037659774, 9.83873536871350434863167217248, 10.38216217993204671401649279225, 11.17213055240634602766644022097, 12.23619290828755751717718436007, 13.42063153950396270044341257401, 13.831850485349030162526729278111, 15.35609017104108104540258759596, 16.13052724330276429249546977330, 17.11274305805915658381674320915, 17.962518728735679357952720170412, 18.60015258526589388469925971282, 19.48215439329094479682909577162, 20.864142234486635539532173436898, 21.53133946531184032720562242431, 22.82411718335490782481199552106, 23.069954266429868154675966534403, 23.75930768020177811594360087519, 25.23760234109843994662664214794

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