Properties

Label 1-19e2-361.96-r0-0-0
Degree $1$
Conductor $361$
Sign $0.883 + 0.468i$
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  + (−0.879 + 0.475i)2-s + (−0.986 − 0.164i)3-s + (0.546 − 0.837i)4-s + (−0.401 − 0.915i)5-s + (0.945 − 0.324i)6-s + (0.546 + 0.837i)7-s + (−0.0825 + 0.996i)8-s + (0.945 + 0.324i)9-s + (0.789 + 0.614i)10-s + (0.945 − 0.324i)11-s + (−0.677 + 0.735i)12-s + (−0.986 − 0.164i)13-s + (−0.879 − 0.475i)14-s + (0.245 + 0.969i)15-s + (−0.401 − 0.915i)16-s + (0.546 + 0.837i)17-s + ⋯
L(s)  = 1  + (−0.879 + 0.475i)2-s + (−0.986 − 0.164i)3-s + (0.546 − 0.837i)4-s + (−0.401 − 0.915i)5-s + (0.945 − 0.324i)6-s + (0.546 + 0.837i)7-s + (−0.0825 + 0.996i)8-s + (0.945 + 0.324i)9-s + (0.789 + 0.614i)10-s + (0.945 − 0.324i)11-s + (−0.677 + 0.735i)12-s + (−0.986 − 0.164i)13-s + (−0.879 − 0.475i)14-s + (0.245 + 0.969i)15-s + (−0.401 − 0.915i)16-s + (0.546 + 0.837i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(361\)    =    \(19^{2}\)
Sign: $0.883 + 0.468i$
Analytic conductor: \(1.67647\)
Root analytic conductor: \(1.67647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{361} (96, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 361,\ (0:\ ),\ 0.883 + 0.468i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5627453424 + 0.1399036629i\)
\(L(\frac12)\) \(\approx\) \(0.5627453424 + 0.1399036629i\)
\(L(1)\) \(\approx\) \(0.5579348647 + 0.06238002869i\)
\(L(1)\) \(\approx\) \(0.5579348647 + 0.06238002869i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
good2 \( 1 + (-0.879 + 0.475i)T \)
3 \( 1 + (-0.986 - 0.164i)T \)
5 \( 1 + (-0.401 - 0.915i)T \)
7 \( 1 + (0.546 + 0.837i)T \)
11 \( 1 + (0.945 - 0.324i)T \)
13 \( 1 + (-0.986 - 0.164i)T \)
17 \( 1 + (0.546 + 0.837i)T \)
23 \( 1 + (-0.986 + 0.164i)T \)
29 \( 1 + (0.546 + 0.837i)T \)
31 \( 1 + (-0.879 - 0.475i)T \)
37 \( 1 + (0.945 - 0.324i)T \)
41 \( 1 + (0.245 + 0.969i)T \)
43 \( 1 + (0.789 - 0.614i)T \)
47 \( 1 + (0.945 - 0.324i)T \)
53 \( 1 + (0.945 - 0.324i)T \)
59 \( 1 + (0.245 + 0.969i)T \)
61 \( 1 + (-0.0825 - 0.996i)T \)
67 \( 1 + (-0.0825 + 0.996i)T \)
71 \( 1 + (-0.0825 + 0.996i)T \)
73 \( 1 + (0.546 + 0.837i)T \)
79 \( 1 + (0.789 - 0.614i)T \)
83 \( 1 + (-0.401 + 0.915i)T \)
89 \( 1 + (0.546 - 0.837i)T \)
97 \( 1 + (-0.0825 - 0.996i)T \)
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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

−24.65852676895453530157069065506, −23.71858573423423479941786557178, −22.662192875806949138377908289501, −22.10239949373770009565940680719, −21.1464696300421564164544065696, −20.07937438856227845675899176603, −19.32056850333100152883646191112, −18.28270276249569094282800303816, −17.647588516930908485106746172651, −16.88733316175096316639192534278, −16.10669144463769545909195433334, −14.94442707695036806877825925957, −13.938135689204278396120715394306, −12.26579450497487517311895841064, −11.79369959807364623121177840345, −10.9384798643565534452474432633, −10.18109322648006576859801325509, −9.42388701691748231105716676212, −7.68679996188867834034268319521, −7.22508867121146761883149903237, −6.26235062701147566040676497601, −4.535445839058179967035037955745, −3.706847861884225892415358771216, −2.1784877477054024422075389238, −0.74372216455021407770964143458, 0.956472915434545624822349064373, 2.002376592013707960434931229376, 4.25806223052128565217192257308, 5.40336284613501685486894627845, 5.955280406988620738272675304154, 7.27364981842711134973734198192, 8.15122670116295779044058513558, 9.08509688188106226039984391296, 10.04163737701323795898280747719, 11.25937259055199366394426537884, 11.947323427733799935561166290233, 12.64444326189804374624683177185, 14.35431362450317523216512935951, 15.23473885956829923403073320825, 16.2199557698024145164360434573, 16.856526686568636789000710808997, 17.54086304593158831525690928163, 18.41908531203626769867573106387, 19.36457750895654437853161604698, 20.07246612221160479308168272114, 21.40095350281071807656792041416, 22.12824738798497367968362672129, 23.46716019024818922012833880541, 24.096484022735141333870308792273, 24.66420796220363277450920201775

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