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Properties

Label 1-335-335.104-r0-0-0
Degree $1$
Conductor $335$
Sign $-0.809 - 0.586i$
Analytic cond. $1.55573$
Root an. cond. $1.55573$
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  + (0.5 − 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + 9-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + 14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s − 3-s + (−0.5 − 0.866i)4-s + (−0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + 9-s + (−0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + 14-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(335\)    =    \(5 \cdot 67\)
Sign: $-0.809 - 0.586i$
Analytic conductor: \(1.55573\)
Root analytic conductor: \(1.55573\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{335} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 335,\ (0:\ ),\ -0.809 - 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2970803573 - 0.9162810220i\)
\(L(\frac12)\) \(\approx\) \(0.2970803573 - 0.9162810220i\)
\(L(1)\) \(\approx\) \(0.7368698712 - 0.5683838906i\)
\(L(1)\) \(\approx\) \(0.7368698712 - 0.5683838906i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
67 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 - T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−25.31862357068259773779137335178, −23.94855398997178378088359062623, −23.579718211918901690489916174425, −23.13974285527416239647612904183, −21.80562795415004848691486356432, −21.37492466407627693577867828844, −20.23668381288107672515677517300, −18.70836808379983018236816452736, −17.8236651922378973791126746525, −17.11552831141999877923285518995, −16.4964335559209303905125264031, −15.4523232413535673030423117958, −14.64166030571179111234510198983, −13.42381299245178008883274373370, −12.79980299430705512497875479835, −11.673103088653062550885278501112, −10.79271417313082336382386542524, −9.63780004382379247432745798827, −8.26298914273329610212873813390, −7.15187141064485138496149376409, −6.63677585476882892149378102398, −5.30410867273772023356923583808, −4.62278179832221924200781549547, −3.645645606678142204769206958815, −1.54351222565612383330001864217, 0.61585183003662569652294307393, 2.05516711100145287309924262935, 3.32155572695209282091632581438, 4.63322076426090065199731999544, 5.623959400891515355085200290011, 6.04051543848089361859872922932, 7.841205855725252648593658792007, 9.04207141412252385716914470086, 10.251520217157926826432718337805, 10.99913147156067151931829289850, 11.74015006968451766536213192107, 12.59978297593158956365366099097, 13.37764586392407899358813247823, 14.63265201891929198803726891407, 15.519580758086080144458507568648, 16.50867952155482529107486654401, 17.76865608222146910536193598969, 18.60150281161746451736966910304, 18.93140413692123504485740203564, 20.70184315818041766324571600007, 21.007345250756131987825726444, 22.065417074517519774280313098347, 22.68822922667450849823559857327, 23.51548109170106392455293104537, 24.36842074275757835587182578972

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