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Properties

Label 1-293-293.60-r0-0-0
Degree $1$
Conductor $293$
Sign $-0.656 + 0.754i$
Analytic cond. $1.36068$
Root an. cond. $1.36068$
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  + (0.651 + 0.758i)2-s + (−0.999 − 0.0430i)3-s + (−0.150 + 0.988i)4-s + (−0.317 − 0.948i)5-s + (−0.618 − 0.785i)6-s + (0.436 + 0.899i)7-s + (−0.847 + 0.530i)8-s + (0.996 + 0.0859i)9-s + (0.512 − 0.858i)10-s + (0.0215 − 0.999i)11-s + (0.192 − 0.981i)12-s + (−0.234 + 0.972i)13-s + (−0.397 + 0.917i)14-s + (0.276 + 0.961i)15-s + (−0.954 − 0.296i)16-s + (0.651 + 0.758i)17-s + ⋯
L(s)  = 1  + (0.651 + 0.758i)2-s + (−0.999 − 0.0430i)3-s + (−0.150 + 0.988i)4-s + (−0.317 − 0.948i)5-s + (−0.618 − 0.785i)6-s + (0.436 + 0.899i)7-s + (−0.847 + 0.530i)8-s + (0.996 + 0.0859i)9-s + (0.512 − 0.858i)10-s + (0.0215 − 0.999i)11-s + (0.192 − 0.981i)12-s + (−0.234 + 0.972i)13-s + (−0.397 + 0.917i)14-s + (0.276 + 0.961i)15-s + (−0.954 − 0.296i)16-s + (0.651 + 0.758i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $-0.656 + 0.754i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (60, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ -0.656 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4191957972 + 0.9206131781i\)
\(L(\frac12)\) \(\approx\) \(0.4191957972 + 0.9206131781i\)
\(L(1)\) \(\approx\) \(0.8220954717 + 0.5418374053i\)
\(L(1)\) \(\approx\) \(0.8220954717 + 0.5418374053i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (0.651 + 0.758i)T \)
3 \( 1 + (-0.999 - 0.0430i)T \)
5 \( 1 + (-0.317 - 0.948i)T \)
7 \( 1 + (0.436 + 0.899i)T \)
11 \( 1 + (0.0215 - 0.999i)T \)
13 \( 1 + (-0.234 + 0.972i)T \)
17 \( 1 + (0.651 + 0.758i)T \)
19 \( 1 + (0.276 + 0.961i)T \)
23 \( 1 + (-0.618 + 0.785i)T \)
29 \( 1 + (-0.474 - 0.880i)T \)
31 \( 1 + (-0.317 + 0.948i)T \)
37 \( 1 + (0.823 + 0.566i)T \)
41 \( 1 + (-0.976 + 0.213i)T \)
43 \( 1 + (0.107 + 0.994i)T \)
47 \( 1 + (0.772 + 0.635i)T \)
53 \( 1 + (0.0215 - 0.999i)T \)
59 \( 1 + (-0.925 + 0.377i)T \)
61 \( 1 + (0.869 + 0.493i)T \)
67 \( 1 + (0.714 + 0.699i)T \)
71 \( 1 + (-0.0645 - 0.997i)T \)
73 \( 1 + (-0.150 - 0.988i)T \)
79 \( 1 + (0.869 + 0.493i)T \)
83 \( 1 + (-0.991 - 0.128i)T \)
89 \( 1 + (0.869 - 0.493i)T \)
97 \( 1 + (-0.683 - 0.729i)T \)
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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

−24.81439820894196194154512931872, −23.66309749789135110561831795488, −23.23206681504003440324382748104, −22.38430933174822280354261177893, −21.91576824450961870787913480992, −20.48451146364025595721399535021, −20.05586044045098613403014942386, −18.593521905862920269870763505217, −18.08431509334258073489297769673, −17.09486616980935877066363299751, −15.69802057550395434585192994711, −14.91975977010507433600384662444, −13.9752718356115839304114777995, −12.84971167821517016630857344008, −11.95982956269125471752449989665, −11.09151949452657148124194071401, −10.42496153630254202490277318273, −9.71713539304256053911340945806, −7.50834392908968926654767562505, −6.850664733478612319030853748, −5.530594391586219473298001359207, −4.60426639572333714535115803272, −3.63225071655813792415633652243, −2.24111360885411626813678160462, −0.63715093251856368582789958537, 1.61884241635387425104489616862, 3.65109460465315054793860413419, 4.674918195428677332238539645503, 5.62076609819987768086597097748, 6.14227451726502937613083802526, 7.666653237253638861118039319480, 8.4429834484236008724868231458, 9.59489590284460100988085037042, 11.40330660730605633015546080233, 11.92364897517630800865877865311, 12.66617769463769320121197450741, 13.73962152412330405534497383057, 14.88943837305896196473924706856, 15.94996093719246442295075929017, 16.5048708575207525630961861773, 17.20699536925686251078554743912, 18.30337344524592871134096587510, 19.22846173983768691771678541781, 20.94566666489293375643849826960, 21.47600356522921663869141511633, 22.17079021094340845855780346024, 23.42431229393643002299242745992, 23.93616121207954250642564263387, 24.54902884801114570321600198702, 25.39833199586780179486130964480

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