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Properties

Label 1-367-367.132-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.807 - 0.589i$
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  + (0.328 − 0.944i)2-s + (−0.935 + 0.352i)3-s + (−0.784 − 0.620i)4-s + (−0.558 − 0.829i)5-s + (0.0257 + 0.999i)6-s + (0.952 − 0.304i)7-s + (−0.843 + 0.536i)8-s + (0.751 − 0.660i)9-s + (−0.967 + 0.254i)10-s + (0.870 + 0.492i)11-s + (0.952 + 0.304i)12-s + (0.751 − 0.660i)13-s + (0.0257 − 0.999i)14-s + (0.815 + 0.579i)15-s + (0.229 + 0.973i)16-s + (0.916 + 0.400i)17-s + ⋯
L(s)  = 1  + (0.328 − 0.944i)2-s + (−0.935 + 0.352i)3-s + (−0.784 − 0.620i)4-s + (−0.558 − 0.829i)5-s + (0.0257 + 0.999i)6-s + (0.952 − 0.304i)7-s + (−0.843 + 0.536i)8-s + (0.751 − 0.660i)9-s + (−0.967 + 0.254i)10-s + (0.870 + 0.492i)11-s + (0.952 + 0.304i)12-s + (0.751 − 0.660i)13-s + (0.0257 − 0.999i)14-s + (0.815 + 0.579i)15-s + (0.229 + 0.973i)16-s + (0.916 + 0.400i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.807 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.807 - 0.589i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.807 - 0.589i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3116725604 - 0.9552887120i\)
\(L(\frac12)\) \(\approx\) \(0.3116725604 - 0.9552887120i\)
\(L(1)\) \(\approx\) \(0.6931129294 - 0.5948910480i\)
\(L(1)\) \(\approx\) \(0.6931129294 - 0.5948910480i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.328 - 0.944i)T \)
3 \( 1 + (-0.935 + 0.352i)T \)
5 \( 1 + (-0.558 - 0.829i)T \)
7 \( 1 + (0.952 - 0.304i)T \)
11 \( 1 + (0.870 + 0.492i)T \)
13 \( 1 + (0.751 - 0.660i)T \)
17 \( 1 + (0.916 + 0.400i)T \)
19 \( 1 + (-0.716 - 0.697i)T \)
23 \( 1 + (0.229 - 0.973i)T \)
29 \( 1 + (-0.894 - 0.447i)T \)
31 \( 1 + (-0.784 - 0.620i)T \)
37 \( 1 + (0.994 - 0.102i)T \)
41 \( 1 + (0.870 - 0.492i)T \)
43 \( 1 + (-0.716 + 0.697i)T \)
47 \( 1 + (-0.988 + 0.153i)T \)
53 \( 1 + (-0.558 - 0.829i)T \)
59 \( 1 + (-0.640 - 0.767i)T \)
61 \( 1 + (0.0257 - 0.999i)T \)
67 \( 1 + (-0.640 + 0.767i)T \)
71 \( 1 + (-0.784 + 0.620i)T \)
73 \( 1 + (0.600 + 0.799i)T \)
79 \( 1 + (0.815 - 0.579i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.988 + 0.153i)T \)
97 \( 1 + (0.229 - 0.973i)T \)
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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

−24.9312523009537539745124390695, −23.81128659836510364605283849435, −23.50932679389345794143729022174, −22.63650433360870171443266114567, −21.76617773217671903939165115628, −21.16319638333296333883589368957, −19.31416873956380843467742844107, −18.453209476979338036347446921852, −18.03086194747281515807769891102, −16.80307469438296448018160673391, −16.34479030914467943295967375217, −15.12757188574554065636388187065, −14.43717533635237899767531699891, −13.575676273591764185059394620281, −12.243021632577518100848061499428, −11.58465352876240243749939717376, −10.80754760040844661358822539470, −9.244568260857198794633790724200, −8.03555173777666641348699198920, −7.315766335112479124676716881456, −6.330149908371150867512887424944, −5.61813122237462000058180493491, −4.41436534595780523515115737918, −3.4677131224934001536990006404, −1.47197165673798952919471849474, 0.72345814987930706715086790379, 1.678297242673048053064402173282, 3.701664471548380070765932723501, 4.38185954015165311830351767993, 5.156635293813744882656887536514, 6.20889350984765948790946327680, 7.83990832453955345609508012661, 8.923756869676740963799160140905, 9.91880936417172690325507183210, 11.10126108845177185550980585440, 11.39824480039197236532105604291, 12.533957546563733378410680031353, 13.01795768977638534811636789463, 14.58617677696937631034256732421, 15.18417870972523055467035535835, 16.54690579086297636121749588101, 17.28612013900346596034165128726, 18.09421213108134211550636024609, 19.15611394590943896439802079707, 20.296089801684382272163629511728, 20.7548901060498774396821487329, 21.58783614851309321242785942176, 22.614032335236892872381755651832, 23.323252990107910642796791868516, 23.91463414430098328510018433945

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