Properties

Label 1-367-367.64-r0-0-0
Degree $1$
Conductor $367$
Sign $0.990 - 0.138i$
Analytic cond. $1.70434$
Root an. cond. $1.70434$
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  + (0.128 − 0.991i)2-s + (0.994 + 0.102i)3-s + (−0.967 − 0.254i)4-s + (0.815 + 0.579i)5-s + (0.229 − 0.973i)6-s + (−0.935 + 0.352i)7-s + (−0.376 + 0.926i)8-s + (0.978 + 0.204i)9-s + (0.679 − 0.733i)10-s + (−0.0771 + 0.997i)11-s + (−0.935 − 0.352i)12-s + (0.978 + 0.204i)13-s + (0.229 + 0.973i)14-s + (0.751 + 0.660i)15-s + (0.870 + 0.492i)16-s + (−0.843 + 0.536i)17-s + ⋯
L(s)  = 1  + (0.128 − 0.991i)2-s + (0.994 + 0.102i)3-s + (−0.967 − 0.254i)4-s + (0.815 + 0.579i)5-s + (0.229 − 0.973i)6-s + (−0.935 + 0.352i)7-s + (−0.376 + 0.926i)8-s + (0.978 + 0.204i)9-s + (0.679 − 0.733i)10-s + (−0.0771 + 0.997i)11-s + (−0.935 − 0.352i)12-s + (0.978 + 0.204i)13-s + (0.229 + 0.973i)14-s + (0.751 + 0.660i)15-s + (0.870 + 0.492i)16-s + (−0.843 + 0.536i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $0.990 - 0.138i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ 0.990 - 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.809371980 - 0.1255054460i\)
\(L(\frac12)\) \(\approx\) \(1.809371980 - 0.1255054460i\)
\(L(1)\) \(\approx\) \(1.437354453 - 0.2801607459i\)
\(L(1)\) \(\approx\) \(1.437354453 - 0.2801607459i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.128 - 0.991i)T \)
3 \( 1 + (0.994 + 0.102i)T \)
5 \( 1 + (0.815 + 0.579i)T \)
7 \( 1 + (-0.935 + 0.352i)T \)
11 \( 1 + (-0.0771 + 0.997i)T \)
13 \( 1 + (0.978 + 0.204i)T \)
17 \( 1 + (-0.843 + 0.536i)T \)
19 \( 1 + (-0.784 + 0.620i)T \)
23 \( 1 + (0.870 - 0.492i)T \)
29 \( 1 + (0.514 - 0.857i)T \)
31 \( 1 + (-0.967 - 0.254i)T \)
37 \( 1 + (0.600 + 0.799i)T \)
41 \( 1 + (-0.0771 - 0.997i)T \)
43 \( 1 + (-0.784 - 0.620i)T \)
47 \( 1 + (-0.179 - 0.983i)T \)
53 \( 1 + (0.815 + 0.579i)T \)
59 \( 1 + (0.0257 + 0.999i)T \)
61 \( 1 + (0.229 + 0.973i)T \)
67 \( 1 + (0.0257 - 0.999i)T \)
71 \( 1 + (-0.967 + 0.254i)T \)
73 \( 1 + (-0.469 - 0.882i)T \)
79 \( 1 + (0.751 - 0.660i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.179 - 0.983i)T \)
97 \( 1 + (0.870 - 0.492i)T \)
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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

−25.00207281676385791428368919883, −23.9604821527841277127838167713, −23.26652818625651293467018176021, −21.89845664778000965162982470209, −21.418587616190744973453791915526, −20.26981459234462567859130907435, −19.35975925288077124717853061812, −18.40079438284992433399080940495, −17.56955254529975926968402905786, −16.28580983640059586787853000019, −16.03640105117322108178129678738, −14.807824642236161150278423688585, −13.82276133488816177544403660354, −13.154574039390224890720309021588, −12.90272381509263641654676621771, −10.81059808651663968144600538208, −9.52054382842041975555670798214, −8.97658774846223241325700099028, −8.24848964282339526956484722661, −6.89468804704713367358207280211, −6.25326429809073909665179968800, −5.01580896409278220717218173609, −3.80887675101261745590488122326, −2.84211326158360792002142164040, −1.02256182719733432973472317052, 1.77061221746773874096320253696, 2.441022659536877558169671171296, 3.46651422790692038219417268844, 4.40166231507082646323291291897, 5.92718548005718064883915602287, 6.94448312292087166209996793025, 8.5642169287757609858069730289, 9.19698842376035532497788034187, 10.1359185221410277895291479206, 10.65861490142814084206286202677, 12.19193770026870054964601808881, 13.264432986906005656842422874222, 13.432132337476188819970358689850, 14.81077017641953012839341671373, 15.23160231757117092400224985317, 16.80050477552315114866525277357, 18.04550112468077686560539102353, 18.688849286985582079040812944547, 19.39528888548297394461247064482, 20.36512078169271210123874778219, 21.06246852670020147800193363989, 21.83997538301989080087231861609, 22.620801161732557792268830807827, 23.49161652661273493399726069799, 24.97506674400177410791605320290

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