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Properties

Label 1-291-291.263-r0-0-0
Degree $1$
Conductor $291$
Sign $0.613 + 0.789i$
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 + (0.980 + 0.195i)5-s + (−0.980 + 0.195i)7-s + (−0.382 − 0.923i)8-s + (−0.831 − 0.555i)10-s + (0.382 + 0.923i)11-s + (−0.195 + 0.980i)13-s + (0.980 + 0.195i)14-s + i·16-s + (0.195 − 0.980i)17-s + (0.980 + 0.195i)19-s + (0.555 + 0.831i)20-s − i·22-s + (−0.831 + 0.555i)23-s + ⋯
L(s)  = 1  + (−0.923 − 0.382i)2-s + (0.707 + 0.707i)4-s + (0.980 + 0.195i)5-s + (−0.980 + 0.195i)7-s + (−0.382 − 0.923i)8-s + (−0.831 − 0.555i)10-s + (0.382 + 0.923i)11-s + (−0.195 + 0.980i)13-s + (0.980 + 0.195i)14-s + i·16-s + (0.195 − 0.980i)17-s + (0.980 + 0.195i)19-s + (0.555 + 0.831i)20-s − i·22-s + (−0.831 + 0.555i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.613 + 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(291\)    =    \(3 \cdot 97\)
Sign: $0.613 + 0.789i$
Analytic conductor: \(1.35139\)
Root analytic conductor: \(1.35139\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{291} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 291,\ (0:\ ),\ 0.613 + 0.789i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7249321641 + 0.3545562675i\)
\(L(\frac12)\) \(\approx\) \(0.7249321641 + 0.3545562675i\)
\(L(1)\) \(\approx\) \(0.7555471527 + 0.08343229631i\)
\(L(1)\) \(\approx\) \(0.7555471527 + 0.08343229631i\)

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 \( 1 + (0.980 + 0.195i)T \)
7 \( 1 + (-0.980 + 0.195i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + (-0.195 + 0.980i)T \)
17 \( 1 + (0.195 - 0.980i)T \)
19 \( 1 + (0.980 + 0.195i)T \)
23 \( 1 + (-0.831 + 0.555i)T \)
29 \( 1 + (-0.831 + 0.555i)T \)
31 \( 1 + (-0.923 + 0.382i)T \)
37 \( 1 + (-0.555 + 0.831i)T \)
41 \( 1 + (-0.555 - 0.831i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.382 - 0.923i)T \)
59 \( 1 + (0.831 + 0.555i)T \)
61 \( 1 + T \)
67 \( 1 + (-0.195 + 0.980i)T \)
71 \( 1 + (-0.555 + 0.831i)T \)
73 \( 1 + (0.707 - 0.707i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (-0.980 - 0.195i)T \)
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

−25.45091020280398892061938232911, −24.663434569971357293279496115952, −23.93266824050795416429036122822, −22.56424337591910754299895786541, −21.807348924333704828109080760904, −20.52860826830132278510633794118, −19.837486030053667764413810508498, −18.86077562258751192529856180935, −18.013246256991275926062700067727, −17.02489068138015488899748190536, −16.48803727996053276977878056099, −15.47950845371978625398255599290, −14.33471306062497734926458804357, −13.399876775083576931022261608772, −12.32651607761937615562279978953, −10.8841130106722584514968995303, −10.083820194560989962403754529387, −9.33347706718807520861013972934, −8.37731998472637580728343245867, −7.17991499882511920270380474125, −6.01088790907242841916645291386, −5.591885702313666948078163806235, −3.494113342620329839597055557539, −2.1947212114058719121269823483, −0.738674969955867410311533038695, 1.53226238220565636162067407334, 2.52907006802329092987851501149, 3.70955235909817987235711668982, 5.47920759362322176397936582429, 6.74645156516752612069525149700, 7.304404837517542337126093218476, 9.062923371004080356087314927239, 9.54688092157011218239652345889, 10.209650883974074890716010838452, 11.5861672746354100270394091050, 12.37923989310247451392138198063, 13.447916653427328352832013218334, 14.51932159518243682217169929357, 15.9100398851581688715928947747, 16.56793541398284115955263399632, 17.57885730590300354137513256243, 18.32614186320450695080756762069, 19.1323359177947954614373982666, 20.1727604593272582805970033578, 20.86925963011404080250813758567, 22.09351571983067440690765423967, 22.41869758416327727673326108347, 24.07606211178624592823245543688, 25.16995714463028871699506550312, 25.68848114955519079935064140399

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