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Properties

Label 1-291-291.20-r0-0-0
Degree $1$
Conductor $291$
Sign $0.944 + 0.329i$
Analytic cond. $1.35139$
Root an. cond. $1.35139$
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  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (0.195 + 0.980i)5-s + (−0.195 + 0.980i)7-s + (0.382 − 0.923i)8-s + (0.555 + 0.831i)10-s + (−0.382 + 0.923i)11-s + (0.980 − 0.195i)13-s + (0.195 + 0.980i)14-s − i·16-s + (−0.980 + 0.195i)17-s + (0.195 + 0.980i)19-s + (0.831 + 0.555i)20-s + i·22-s + (0.555 − 0.831i)23-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (0.707 − 0.707i)4-s + (0.195 + 0.980i)5-s + (−0.195 + 0.980i)7-s + (0.382 − 0.923i)8-s + (0.555 + 0.831i)10-s + (−0.382 + 0.923i)11-s + (0.980 − 0.195i)13-s + (0.195 + 0.980i)14-s − i·16-s + (−0.980 + 0.195i)17-s + (0.195 + 0.980i)19-s + (0.831 + 0.555i)20-s + i·22-s + (0.555 − 0.831i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(291\)    =    \(3 \cdot 97\)
Sign: $0.944 + 0.329i$
Analytic conductor: \(1.35139\)
Root analytic conductor: \(1.35139\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{291} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 291,\ (0:\ ),\ 0.944 + 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.147700671 + 0.3639375138i\)
\(L(\frac12)\) \(\approx\) \(2.147700671 + 0.3639375138i\)
\(L(1)\) \(\approx\) \(1.781081877 + 0.05873831381i\)
\(L(1)\) \(\approx\) \(1.781081877 + 0.05873831381i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
97 \( 1 \)
good2 \( 1 + (0.923 - 0.382i)T \)
5 \( 1 + (0.195 + 0.980i)T \)
7 \( 1 + (-0.195 + 0.980i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + (0.980 - 0.195i)T \)
17 \( 1 + (-0.980 + 0.195i)T \)
19 \( 1 + (0.195 + 0.980i)T \)
23 \( 1 + (0.555 - 0.831i)T \)
29 \( 1 + (0.555 - 0.831i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (-0.831 + 0.555i)T \)
41 \( 1 + (-0.831 - 0.555i)T \)
43 \( 1 + (0.707 - 0.707i)T \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (-0.382 - 0.923i)T \)
59 \( 1 + (-0.555 - 0.831i)T \)
61 \( 1 + T \)
67 \( 1 + (0.980 - 0.195i)T \)
71 \( 1 + (-0.831 + 0.555i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + (-0.923 - 0.382i)T \)
83 \( 1 + (-0.195 - 0.980i)T \)
89 \( 1 + (-0.382 + 0.923i)T \)
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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

−25.26895802353328581657882963052, −24.21586371036146418365092426581, −23.79073978130879473727245881455, −22.932049090378539196059192027, −21.77079208953918902653192498858, −21.029144861224413878527116419938, −20.26027498179079593020353716913, −19.4358330649953641628425819020, −17.76454893534974550708576569292, −16.99316786654380780901436667884, −16.04776258663928745912541645010, −15.59995512790885546641212571473, −13.8800959869535166379328840472, −13.54876175112842804818554561584, −12.79839686922971424329103115083, −11.46527988044863264583707680100, −10.749883761823342034658806790736, −9.11577606697039169710147694384, −8.22192878083081377424128144937, −7.04559345272942604424158471786, −6.04751622240710095284311793445, −4.95728782623233351785083886462, −4.0843155635375339312961067634, −2.931767871864101622999780192741, −1.199961060953171477237991872296, 1.918027302268660046682236492393, 2.736669296271188204322402227905, 3.85351865074499861066135340440, 5.169303755524377736653240252, 6.21238480528202024046965774004, 6.89858297744233260342204594352, 8.43254683062780417673752485960, 9.899864285872657152848728517885, 10.602926386161521997132254363578, 11.64576344465548617017556784389, 12.51479180560057775501188160593, 13.48094522825215455917810230199, 14.41899115233044816575418457133, 15.41880195786435002275197185700, 15.72420664316861751427064839193, 17.495691293354150978094297828480, 18.5782313506350837184325915174, 19.05061876030965325130911548073, 20.40992503935238144954495103800, 21.08981439556606223404933822867, 22.10240269756909304036925568289, 22.71677242375174779702748275379, 23.3567156987689153924532785083, 24.70353939220354391895526751028, 25.32820765789144682120793067381

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