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Properties

Label 1-19e2-361.161-r0-0-0
Degree $1$
Conductor $361$
Sign $-0.670 - 0.742i$
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  + (0.663 − 0.748i)2-s + (−0.333 + 0.942i)3-s + (−0.119 − 0.992i)4-s + (0.280 − 0.959i)5-s + (0.484 + 0.875i)6-s + (0.451 − 0.892i)7-s + (−0.821 − 0.569i)8-s + (−0.777 − 0.628i)9-s + (−0.531 − 0.847i)10-s + (−0.191 + 0.981i)11-s + (0.975 + 0.218i)12-s + (−0.00918 − 0.999i)13-s + (−0.367 − 0.929i)14-s + (0.811 + 0.584i)15-s + (−0.971 + 0.236i)16-s + (−0.800 − 0.599i)17-s + ⋯
L(s)  = 1  + (0.663 − 0.748i)2-s + (−0.333 + 0.942i)3-s + (−0.119 − 0.992i)4-s + (0.280 − 0.959i)5-s + (0.484 + 0.875i)6-s + (0.451 − 0.892i)7-s + (−0.821 − 0.569i)8-s + (−0.777 − 0.628i)9-s + (−0.531 − 0.847i)10-s + (−0.191 + 0.981i)11-s + (0.975 + 0.218i)12-s + (−0.00918 − 0.999i)13-s + (−0.367 − 0.929i)14-s + (0.811 + 0.584i)15-s + (−0.971 + 0.236i)16-s + (−0.800 − 0.599i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $-0.670 - 0.742i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ -0.670 - 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5599435512 - 1.259948247i\)
\(L(\frac12)\) \(\approx\) \(0.5599435512 - 1.259948247i\)
\(L(1)\) \(\approx\) \(1.029115012 - 0.6824588367i\)
\(L(1)\) \(\approx\) \(1.029115012 - 0.6824588367i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (0.663 - 0.748i)T \)
3 \( 1 + (-0.333 + 0.942i)T \)
5 \( 1 + (0.280 - 0.959i)T \)
7 \( 1 + (0.451 - 0.892i)T \)
11 \( 1 + (-0.191 + 0.981i)T \)
13 \( 1 + (-0.00918 - 0.999i)T \)
17 \( 1 + (-0.800 - 0.599i)T \)
23 \( 1 + (-0.649 + 0.760i)T \)
29 \( 1 + (-0.227 - 0.973i)T \)
31 \( 1 + (0.0275 + 0.999i)T \)
37 \( 1 + (0.945 - 0.324i)T \)
41 \( 1 + (0.997 - 0.0734i)T \)
43 \( 1 + (0.999 + 0.0367i)T \)
47 \( 1 + (-0.999 - 0.0183i)T \)
53 \( 1 + (0.515 - 0.856i)T \)
59 \( 1 + (-0.562 - 0.826i)T \)
61 \( 1 + (-0.703 - 0.710i)T \)
67 \( 1 + (0.418 - 0.908i)T \)
71 \( 1 + (-0.703 + 0.710i)T \)
73 \( 1 + (0.919 - 0.393i)T \)
79 \( 1 + (-0.467 - 0.883i)T \)
83 \( 1 + (0.993 - 0.110i)T \)
89 \( 1 + (0.919 + 0.393i)T \)
97 \( 1 + (0.418 + 0.908i)T \)
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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

−24.675174286849771510997941524847, −24.259914016944692519537785993327, −23.500052305273161236914143502614, −22.34042229118204251499638217603, −21.94182789367291150319532998910, −21.09134970984145094764685478604, −19.49435916678372548132542338621, −18.44246253744626505194876951485, −18.14023822218836075263063253313, −17.03533319465190951504624361398, −16.15176918544898022851800041395, −14.96021704859900789447381346253, −14.26196639657727689217899192509, −13.52191767510635930982474860080, −12.554301500480214183897810536840, −11.53481991580100545426926179247, −10.97832501077045389064508407268, −9.03814167835210075146374490455, −8.170961810339952380275744339988, −7.19386761174175615042506085818, −6.12809288320651148725647059743, −5.880121614364723264457325254486, −4.39555129096752230488932493121, −2.86639205494756404080273140014, −2.063323208314551332921772741768, 0.683950657563256973341407717869, 2.14703254199846514467924671023, 3.64643506897826089954777503042, 4.591900855990147143468794238784, 5.06848503029678589527972429758, 6.175900170788810966777189685030, 7.805711127106707593280571398976, 9.24443173302851598700044919872, 9.903772197741503493074873534816, 10.74075884605388058318667963009, 11.643632795590951450854407742326, 12.631245592269237466503902807658, 13.48366445554764908846215423669, 14.44752690779900213385844743873, 15.45320446336825776897924610907, 16.1655973695111128142071601888, 17.572780037366421268514927413601, 17.787119651116036289349904334283, 19.86161187333879526718750246897, 20.14840122819849322515298025093, 20.90100090932751807417962444776, 21.57280799974053764627788389657, 22.730321174615781313953455253088, 23.16713873130681326029566892176, 24.16747538325360503292512394665

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