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Properties

Label 1-367-367.74-r0-0-0
Degree $1$
Conductor $367$
Sign $-0.432 + 0.901i$
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  + (0.423 − 0.905i)2-s + (−0.558 − 0.829i)3-s + (−0.640 − 0.767i)4-s + (0.916 − 0.400i)5-s + (−0.988 + 0.153i)6-s + (−0.279 − 0.960i)7-s + (−0.967 + 0.254i)8-s + (−0.376 + 0.926i)9-s + (0.0257 − 0.999i)10-s + (−0.998 + 0.0514i)11-s + (−0.279 + 0.960i)12-s + (−0.376 + 0.926i)13-s + (−0.988 − 0.153i)14-s + (−0.843 − 0.536i)15-s + (−0.179 + 0.983i)16-s + (−0.784 + 0.620i)17-s + ⋯
L(s)  = 1  + (0.423 − 0.905i)2-s + (−0.558 − 0.829i)3-s + (−0.640 − 0.767i)4-s + (0.916 − 0.400i)5-s + (−0.988 + 0.153i)6-s + (−0.279 − 0.960i)7-s + (−0.967 + 0.254i)8-s + (−0.376 + 0.926i)9-s + (0.0257 − 0.999i)10-s + (−0.998 + 0.0514i)11-s + (−0.279 + 0.960i)12-s + (−0.376 + 0.926i)13-s + (−0.988 − 0.153i)14-s + (−0.843 − 0.536i)15-s + (−0.179 + 0.983i)16-s + (−0.784 + 0.620i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(367\)
Sign: $-0.432 + 0.901i$
Analytic conductor: \(1.70434\)
Root analytic conductor: \(1.70434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{367} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 367,\ (0:\ ),\ -0.432 + 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3890304113 - 0.6181751472i\)
\(L(\frac12)\) \(\approx\) \(-0.3890304113 - 0.6181751472i\)
\(L(1)\) \(\approx\) \(0.4434962421 - 0.7452492403i\)
\(L(1)\) \(\approx\) \(0.4434962421 - 0.7452492403i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad367 \( 1 \)
good2 \( 1 + (0.423 - 0.905i)T \)
3 \( 1 + (-0.558 - 0.829i)T \)
5 \( 1 + (0.916 - 0.400i)T \)
7 \( 1 + (-0.279 - 0.960i)T \)
11 \( 1 + (-0.998 + 0.0514i)T \)
13 \( 1 + (-0.376 + 0.926i)T \)
17 \( 1 + (-0.784 + 0.620i)T \)
19 \( 1 + (-0.0771 - 0.997i)T \)
23 \( 1 + (-0.179 - 0.983i)T \)
29 \( 1 + (-0.935 + 0.352i)T \)
31 \( 1 + (-0.640 - 0.767i)T \)
37 \( 1 + (0.815 - 0.579i)T \)
41 \( 1 + (-0.998 - 0.0514i)T \)
43 \( 1 + (-0.0771 + 0.997i)T \)
47 \( 1 + (0.600 - 0.799i)T \)
53 \( 1 + (0.916 - 0.400i)T \)
59 \( 1 + (0.514 - 0.857i)T \)
61 \( 1 + (-0.988 - 0.153i)T \)
67 \( 1 + (0.514 + 0.857i)T \)
71 \( 1 + (-0.640 + 0.767i)T \)
73 \( 1 + (0.751 - 0.660i)T \)
79 \( 1 + (-0.843 + 0.536i)T \)
83 \( 1 + T \)
89 \( 1 + (0.600 - 0.799i)T \)
97 \( 1 + (-0.179 - 0.983i)T \)
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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

−25.39839934462747456772437046656, −24.471929587002053965000922415918, −23.346263705879838869196620375692, −22.497753275737819037957763336223, −21.99234577508149203364693786341, −21.29757197983065916128819312327, −20.43136722722857583738021429487, −18.5082172175601628060182275734, −18.05180801443593514073332524767, −17.18056242336291283772314780063, −16.25219113788680305961060268517, −15.354809975138177966654713238085, −14.95156349751145739266209916200, −13.70175218360560145467403918254, −12.828173548595433242274137579743, −11.855013738716217452452480284048, −10.554138079089517795833990453418, −9.65910835390875839242514037660, −8.86683521199135025479365433618, −7.56190172576919472006646265436, −6.28675172927241627047030103172, −5.53998705785942367525224447662, −5.08513235630181955098210967832, −3.507662997794837277718704980638, −2.54651997407242489699617674364, 0.38523127796195604892057399301, 1.787840380631932021797259904199, 2.51917737998751355654957514968, 4.27187319160746117121491986852, 5.13325139158011415460424131608, 6.18276821542426332262675541233, 7.08537903351691912477565587801, 8.56949488372374000020347294658, 9.69737632613409357367894993321, 10.636115444344448728418921366139, 11.29604348497092824980295928368, 12.61960252087509304913692376964, 13.15974465193508296179452056871, 13.66771366138725510051429331258, 14.72971598814864764783110738859, 16.3767645805483302377558998878, 17.18386451866978488300836339995, 18.03114842380580348776832373601, 18.77192782138245224804296687961, 19.844211410803748571686589083847, 20.45149527196099199820913502826, 21.61710645363860868127419180689, 22.16951650847138391286817919085, 23.25684233342061723304937594551, 23.98424342631144083519845447388

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