Properties

Label 1-19e2-361.58-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.596 + 0.802i$
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  + (0.945 + 0.324i)2-s + (−0.401 + 0.915i)3-s + (0.789 + 0.614i)4-s + (0.245 + 0.969i)5-s + (−0.677 + 0.735i)6-s + (0.789 − 0.614i)7-s + (0.546 + 0.837i)8-s + (−0.677 − 0.735i)9-s + (−0.0825 + 0.996i)10-s + (−0.677 + 0.735i)11-s + (−0.879 + 0.475i)12-s + (−0.401 + 0.915i)13-s + (0.945 − 0.324i)14-s + (−0.986 − 0.164i)15-s + (0.245 + 0.969i)16-s + (0.789 − 0.614i)17-s + ⋯
L(s)  = 1  + (0.945 + 0.324i)2-s + (−0.401 + 0.915i)3-s + (0.789 + 0.614i)4-s + (0.245 + 0.969i)5-s + (−0.677 + 0.735i)6-s + (0.789 − 0.614i)7-s + (0.546 + 0.837i)8-s + (−0.677 − 0.735i)9-s + (−0.0825 + 0.996i)10-s + (−0.677 + 0.735i)11-s + (−0.879 + 0.475i)12-s + (−0.401 + 0.915i)13-s + (0.945 − 0.324i)14-s + (−0.986 − 0.164i)15-s + (0.245 + 0.969i)16-s + (0.789 − 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.596 + 0.802i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.596 + 0.802i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9573099623 + 1.905395156i\)
\(L(\frac12)\) \(\approx\) \(0.9573099623 + 1.905395156i\)
\(L(1)\) \(\approx\) \(1.314097321 + 1.085485995i\)
\(L(1)\) \(\approx\) \(1.314097321 + 1.085485995i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.945 + 0.324i)T \)
3 \( 1 + (-0.401 + 0.915i)T \)
5 \( 1 + (0.245 + 0.969i)T \)
7 \( 1 + (0.789 - 0.614i)T \)
11 \( 1 + (-0.677 + 0.735i)T \)
13 \( 1 + (-0.401 + 0.915i)T \)
17 \( 1 + (0.789 - 0.614i)T \)
23 \( 1 + (-0.401 - 0.915i)T \)
29 \( 1 + (0.789 - 0.614i)T \)
31 \( 1 + (0.945 - 0.324i)T \)
37 \( 1 + (-0.677 + 0.735i)T \)
41 \( 1 + (-0.986 - 0.164i)T \)
43 \( 1 + (-0.0825 - 0.996i)T \)
47 \( 1 + (-0.677 + 0.735i)T \)
53 \( 1 + (-0.677 + 0.735i)T \)
59 \( 1 + (-0.986 - 0.164i)T \)
61 \( 1 + (0.546 - 0.837i)T \)
67 \( 1 + (0.546 + 0.837i)T \)
71 \( 1 + (0.546 + 0.837i)T \)
73 \( 1 + (0.789 - 0.614i)T \)
79 \( 1 + (-0.0825 - 0.996i)T \)
83 \( 1 + (0.245 - 0.969i)T \)
89 \( 1 + (0.789 + 0.614i)T \)
97 \( 1 + (0.546 - 0.837i)T \)
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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

−24.22097794081637696254388440567, −23.71234915580540899392892660904, −22.82036189616012350613009007396, −21.56216512946603990558880214909, −21.2316227384842700271962015839, −20.03167377886204972672599515843, −19.33010223741586292060536261385, −18.23555083999707234695838611114, −17.37758551761727151060883548567, −16.33069052829049575876551483288, −15.39135955563022943259065100918, −14.212143986095941220191592863314, −13.45232146002272314318466864285, −12.56246269834356708379229878961, −12.0749529222131765225626666855, −11.09996916491519455340647024700, −10.06088517195755382507139769726, −8.38440890415317316928684204194, −7.7851378744670702807169185791, −6.256286271800939526002978173898, −5.35931616546937660021590032155, −5.020118943825468379246642297481, −3.211353360591242591008730926251, −2.01646290345405726885886219072, −1.06084284250996130059423979455, 2.15447390044491569301603856565, 3.26197482875129472160650779504, 4.45034971620076127760393275104, 5.00291915222878479239122035990, 6.26998088387339972512619255432, 7.12015045031898150756544787160, 8.1540961848407321005624953332, 9.89630433777863898400767582694, 10.51546923525657629965653540647, 11.50917304714069221694889286988, 12.168380230071949486299816154089, 13.81057542171986315570640034905, 14.2858856083950229913753569001, 15.11161538312784284840263836171, 15.88929797582037706038607037588, 16.99366393397615846599844304661, 17.55098434868021097977549585249, 18.76377067379077245437361699689, 20.32126377720278905879902406324, 20.889022096088349752589257718796, 21.6216885866618744571932385962, 22.46712920446869042548220546861, 23.18243699242862122663903917249, 23.77728671738657854528672410111, 24.99668018856508961100436633972

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