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Properties

Label 1-293-293.31-r0-0-0
Degree $1$
Conductor $293$
Sign $-0.977 + 0.211i$
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  + (0.548 + 0.835i)2-s + (−0.107 − 0.994i)3-s + (−0.397 + 0.917i)4-s + (−0.0215 + 0.999i)5-s + (0.772 − 0.635i)6-s + (−0.941 − 0.337i)7-s + (−0.985 + 0.171i)8-s + (−0.976 + 0.213i)9-s + (−0.847 + 0.530i)10-s + (0.744 + 0.668i)11-s + (0.954 + 0.296i)12-s + (−0.192 + 0.981i)13-s + (−0.234 − 0.972i)14-s + (0.996 − 0.0859i)15-s + (−0.683 − 0.729i)16-s + (−0.548 − 0.835i)17-s + ⋯
L(s)  = 1  + (0.548 + 0.835i)2-s + (−0.107 − 0.994i)3-s + (−0.397 + 0.917i)4-s + (−0.0215 + 0.999i)5-s + (0.772 − 0.635i)6-s + (−0.941 − 0.337i)7-s + (−0.985 + 0.171i)8-s + (−0.976 + 0.213i)9-s + (−0.847 + 0.530i)10-s + (0.744 + 0.668i)11-s + (0.954 + 0.296i)12-s + (−0.192 + 0.981i)13-s + (−0.234 − 0.972i)14-s + (0.996 − 0.0859i)15-s + (−0.683 − 0.729i)16-s + (−0.548 − 0.835i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 293 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(293\)
Sign: $-0.977 + 0.211i$
Analytic conductor: \(1.36068\)
Root analytic conductor: \(1.36068\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{293} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 293,\ (0:\ ),\ -0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07688773302 + 0.7171485712i\)
\(L(\frac12)\) \(\approx\) \(0.07688773302 + 0.7171485712i\)
\(L(1)\) \(\approx\) \(0.7672742662 + 0.4737769050i\)
\(L(1)\) \(\approx\) \(0.7672742662 + 0.4737769050i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad293 \( 1 \)
good2 \( 1 + (0.548 + 0.835i)T \)
3 \( 1 + (-0.107 - 0.994i)T \)
5 \( 1 + (-0.0215 + 0.999i)T \)
7 \( 1 + (-0.941 - 0.337i)T \)
11 \( 1 + (0.744 + 0.668i)T \)
13 \( 1 + (-0.192 + 0.981i)T \)
17 \( 1 + (-0.548 - 0.835i)T \)
19 \( 1 + (-0.996 + 0.0859i)T \)
23 \( 1 + (-0.772 - 0.635i)T \)
29 \( 1 + (-0.436 + 0.899i)T \)
31 \( 1 + (0.0215 + 0.999i)T \)
37 \( 1 + (-0.0645 + 0.997i)T \)
41 \( 1 + (-0.512 + 0.858i)T \)
43 \( 1 + (0.869 - 0.493i)T \)
47 \( 1 + (0.150 + 0.988i)T \)
53 \( 1 + (-0.744 - 0.668i)T \)
59 \( 1 + (0.823 - 0.566i)T \)
61 \( 1 + (0.276 - 0.961i)T \)
67 \( 1 + (0.357 + 0.933i)T \)
71 \( 1 + (0.584 + 0.811i)T \)
73 \( 1 + (-0.397 - 0.917i)T \)
79 \( 1 + (-0.276 + 0.961i)T \)
83 \( 1 + (-0.317 - 0.948i)T \)
89 \( 1 + (-0.276 - 0.961i)T \)
97 \( 1 + (-0.890 + 0.455i)T \)
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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

−24.91456931664331344145249178682, −23.96336044206690631083696928918, −22.90076067789786126531926189229, −22.13653054118306167863868156015, −21.525358078255413977525324233140, −20.609280450101278459362801451032, −19.72225351011587041747792710814, −19.27699816248495044040359327807, −17.588036975150476161833282075975, −16.738540407385400655583816089603, −15.63586572427073044942046296367, −15.05762632750383029582245109377, −13.72002839603231115945642489980, −12.85009202316101471080157608682, −12.021593984356668988585417452212, −11.002411229781382252030669142681, −9.99372331403029197828205562398, −9.218093638071880752204569005948, −8.44075255248865483437400989023, −6.047334638311546087906343484718, −5.62173170419368307929702379399, −4.20101822281362086466805323976, −3.67120967108071359296318022173, −2.272319936217250678652949328579, −0.37654827800767908958651195365, 2.17809266604020950201727054354, 3.31781453929337051600237574323, 4.523310032919212463832154167709, 6.16998159421504071746560085329, 6.7912245345813323117705678819, 7.17028118201047340246045173865, 8.56743195232835175002406054186, 9.715810129997787739913884328773, 11.24587607022363061988587825434, 12.205016800950713946145504850161, 13.01127936975552601513743400911, 14.12425624913791517032690631393, 14.45891353024476785216726035606, 15.78295444401502251008740920212, 16.777624915098462607407851730955, 17.59322108171559645766931733617, 18.51701837043685518719685849902, 19.30688572441231138285371927342, 20.362180484509155889623979174116, 22.02432730916769010747843490678, 22.40946871238690273416537314814, 23.28491253902410014825070182643, 23.91090769000772306839680485566, 25.08078333008915711106851371827, 25.675818088385503278328627234421

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