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Properties

Label 1-19e2-361.6-r0-0-0
Degree $1$
Conductor $361$
Sign $0.480 - 0.876i$
Analytic cond. $1.67647$
Root an. cond. $1.67647$
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  + (−0.842 + 0.539i)2-s + (0.811 − 0.584i)3-s + (0.418 − 0.908i)4-s + (0.983 − 0.182i)5-s + (−0.367 + 0.929i)6-s + (−0.821 + 0.569i)7-s + (0.137 + 0.990i)8-s + (0.315 − 0.948i)9-s + (−0.729 + 0.684i)10-s + (0.0275 − 0.999i)11-s + (−0.191 − 0.981i)12-s + (−0.435 − 0.900i)13-s + (0.384 − 0.922i)14-s + (0.690 − 0.723i)15-s + (−0.649 − 0.760i)16-s + (−0.995 − 0.0917i)17-s + ⋯
L(s)  = 1  + (−0.842 + 0.539i)2-s + (0.811 − 0.584i)3-s + (0.418 − 0.908i)4-s + (0.983 − 0.182i)5-s + (−0.367 + 0.929i)6-s + (−0.821 + 0.569i)7-s + (0.137 + 0.990i)8-s + (0.315 − 0.948i)9-s + (−0.729 + 0.684i)10-s + (0.0275 − 0.999i)11-s + (−0.191 − 0.981i)12-s + (−0.435 − 0.900i)13-s + (0.384 − 0.922i)14-s + (0.690 − 0.723i)15-s + (−0.649 − 0.760i)16-s + (−0.995 − 0.0917i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.480 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.480 - 0.876i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.480 - 0.876i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9789887466 - 0.5798287238i\)
\(L(\frac12)\) \(\approx\) \(0.9789887466 - 0.5798287238i\)
\(L(1)\) \(\approx\) \(0.9642956520 - 0.1835542466i\)
\(L(1)\) \(\approx\) \(0.9642956520 - 0.1835542466i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.842 + 0.539i)T \)
3 \( 1 + (0.811 - 0.584i)T \)
5 \( 1 + (0.983 - 0.182i)T \)
7 \( 1 + (-0.821 + 0.569i)T \)
11 \( 1 + (0.0275 - 0.999i)T \)
13 \( 1 + (-0.435 - 0.900i)T \)
17 \( 1 + (-0.995 - 0.0917i)T \)
23 \( 1 + (0.100 - 0.994i)T \)
29 \( 1 + (0.967 - 0.254i)T \)
31 \( 1 + (0.975 + 0.218i)T \)
37 \( 1 + (-0.879 + 0.475i)T \)
41 \( 1 + (-0.896 + 0.443i)T \)
43 \( 1 + (-0.227 + 0.973i)T \)
47 \( 1 + (-0.621 - 0.783i)T \)
53 \( 1 + (0.989 - 0.146i)T \)
59 \( 1 + (0.832 + 0.554i)T \)
61 \( 1 + (-0.531 - 0.847i)T \)
67 \( 1 + (0.741 + 0.670i)T \)
71 \( 1 + (-0.531 + 0.847i)T \)
73 \( 1 + (0.577 - 0.816i)T \)
79 \( 1 + (0.957 - 0.289i)T \)
83 \( 1 + (0.635 + 0.771i)T \)
89 \( 1 + (0.577 + 0.816i)T \)
97 \( 1 + (0.741 - 0.670i)T \)
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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

−25.41230202837490096548207250383, −24.390070580054642012168356514530, −22.72339726724299352259401129700, −21.916770749531536012317957825851, −21.21926111025465630166192393117, −20.36700495156283647347128981716, −19.63786404244018658480802155968, −18.95410846745938130779826593520, −17.6950642135030432158186409372, −17.07927454004957029896343281223, −16.07959542062513011602703826062, −15.19923336107076923995340147846, −13.8689494352231228159533900294, −13.299904393099472620519842237487, −12.16650930262745040641583591459, −10.75575140997471828693000604407, −10.00605274713165230948525072770, −9.48676987621018711245288500679, −8.700167764000056431011993159184, −7.26959452361890855459556758466, −6.645789631539945767421768763230, −4.73726442210449603816460912612, −3.66105512039269864907395237063, −2.52527650505955210832851223246, −1.7397567923338621325100868179, 0.81841182735364431042484314851, 2.26787367885580208495648181815, 2.98218655760658230892040980694, 5.12116400998320193787906279268, 6.32765589119742701810952350819, 6.69003874483192915373598550318, 8.2969395632227685809105212855, 8.69688915299069752127728485687, 9.7032004248244751352165287047, 10.41789980271956761234856723875, 11.97080596419448709509020511363, 13.13934263448572516425300378433, 13.77654496436837833674790511809, 14.83095636479628434589182443259, 15.66270489945580322845091705220, 16.645876157145740838119379616830, 17.69659598318610116734279386491, 18.315263445176718412618759915840, 19.20478676958201212481176929020, 19.86724782057708213786887523642, 20.813986080927066250040912313971, 21.88140968433672797718992465933, 22.97697545247285935626285397432, 24.35296683981829292227695164852, 24.76398596844491423703959877119

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