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Properties

Label 1-335-335.297-r1-0-0
Degree $1$
Conductor $335$
Sign $0.0734 + 0.997i$
Analytic cond. $36.0007$
Root an. cond. $36.0007$
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  + (0.866 − 0.5i)2-s − i·3-s + (0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s + (−0.866 − 0.5i)7-s − i·8-s − 9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)12-s + (−0.866 + 0.5i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s − i·3-s + (0.5 − 0.866i)4-s + (−0.5 − 0.866i)6-s + (−0.866 − 0.5i)7-s − i·8-s − 9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)12-s + (−0.866 + 0.5i)13-s − 14-s + (−0.5 − 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0734 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 335 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0734 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(335\)    =    \(5 \cdot 67\)
Sign: $0.0734 + 0.997i$
Analytic conductor: \(36.0007\)
Root analytic conductor: \(36.0007\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{335} (297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 335,\ (1:\ ),\ 0.0734 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2655222892 - 0.2466935938i\)
\(L(\frac12)\) \(\approx\) \(-0.2655222892 - 0.2466935938i\)
\(L(1)\) \(\approx\) \(0.8849387179 - 0.7535796206i\)
\(L(1)\) \(\approx\) \(0.8849387179 - 0.7535796206i\)

Euler product

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

−25.56981769888083918222637800886, −24.46651837930200783421093530943, −23.59785708161618980454165794503, −22.52508277575010295132071580155, −21.98648649861181945909294657681, −21.41018040267404481104360203734, −20.310709997612724586608618547312, −19.52153015862714136679686711698, −18.11020491310400073400263102291, −16.83769794020971017666355974663, −16.2965889667782229439995079838, −15.4912134285599297421363941926, −14.76018537898337759554783234321, −13.794884923413573225200457447901, −12.75112032380540070746453901921, −11.874527376011801554967882432098, −10.77301830321228031332831843123, −9.75547907367985784290358912050, −8.64291252557706802239383350020, −7.64083191218538936367154122837, −6.167106264252945826882643872407, −5.53617156959429585990970609281, −4.531176776777206551938274264480, −3.24530356915403101645429090085, −2.749336734751660548988219587854, 0.0711862343390026995497202466, 1.505367747095589922789543812924, 2.57966463683747258314383099694, 3.617933978032444516260658705858, 4.98649796443086901354659030119, 6.03149007222235529849705799108, 7.02910178885119893921208106815, 7.72262614606043706159124918321, 9.61171970775244655838946342257, 10.1953117016582381238648419702, 11.67864980495089469799662378267, 12.27973952805117041374998622648, 13.031880713612882600486776764667, 13.94335409281064011612519767471, 14.59426987018378213822087411673, 15.90388898233541928695153389866, 16.82421256100995100248240185361, 18.107001872885592922790114697860, 18.9305401882367284494166156579, 19.80167004840048320077677053300, 20.31619009413813843357821709587, 21.49634501441244959440834019015, 22.59916935280043199697741011767, 23.197085948600545302917955957375, 23.77126422066700245119325547211

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