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Properties

Label 1-1148-1148.331-r0-0-0
Degree $1$
Conductor $1148$
Sign $-0.715 - 0.698i$
Analytic cond. $5.33128$
Root an. cond. $5.33128$
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  + (−0.965 + 0.258i)3-s + (0.866 − 0.5i)5-s + (0.866 − 0.5i)9-s + (0.965 − 0.258i)11-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)15-s + (−0.258 − 0.965i)17-s + (−0.965 − 0.258i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (−0.707 + 0.707i)27-s + (−0.707 − 0.707i)29-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.5i)33-s + (−0.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (0.866 − 0.5i)5-s + (0.866 − 0.5i)9-s + (0.965 − 0.258i)11-s + (−0.707 − 0.707i)13-s + (−0.707 + 0.707i)15-s + (−0.258 − 0.965i)17-s + (−0.965 − 0.258i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s + (−0.707 + 0.707i)27-s + (−0.707 − 0.707i)29-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.5i)33-s + (−0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.715 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.715 - 0.698i$
Analytic conductor: \(5.33128\)
Root analytic conductor: \(5.33128\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1148,\ (0:\ ),\ -0.715 - 0.698i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2733659724 - 0.6710861527i\)
\(L(\frac12)\) \(\approx\) \(0.2733659724 - 0.6710861527i\)
\(L(1)\) \(\approx\) \(0.7709358110 - 0.1964674800i\)
\(L(1)\) \(\approx\) \(0.7709358110 - 0.1964674800i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (-0.258 - 0.965i)T \)
19 \( 1 + (-0.965 - 0.258i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.965 - 0.258i)T \)
53 \( 1 + (-0.965 + 0.258i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.866 + 0.5i)T \)
67 \( 1 + (0.258 + 0.965i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.258 + 0.965i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−21.78147271037125378361605140627, −21.15200456020299610773003211764, −19.8998335035744478152103146712, −19.09216049425123477779342119862, −18.48474048335708530294755056941, −17.48907043343088392816088721324, −17.12175378156389330552373058448, −16.55457593805385880978326384170, −15.25801764533848321037123972175, −14.6173684067097375932849235574, −13.73157641264398408634517563915, −12.89663681847067871223348737241, −12.15832338660785138626153032539, −11.32713711192722393832380945135, −10.59687133258294241961330660778, −9.793869458198405971092027438160, −9.09432699084069059470910419502, −7.758274517972533953181516395154, −6.78138977139908511455541425090, −6.352674180767606692861857611428, −5.515721860957067188407092907730, −4.52634777425969446505707759369, −3.59492861540254239569996301468, −1.92235209684741144487020364391, −1.688111264255857320419568397485, 0.32148589067993486613820489134, 1.48126144743609286348887927133, 2.57116060399393773044443727251, 3.97441667704776897798634025538, 4.81281746887292761693141843900, 5.51913184489825255755943668876, 6.3640157715194849690718134481, 7.012257089263308753174519584808, 8.333957129972386392495664127357, 9.32064100059084031963858014541, 9.84718645635823439501926114100, 10.75433120528921012745583236893, 11.51317935161488268384248124237, 12.49872650030538473330038362880, 12.88335821836946530647523196510, 14.03616787417211636165437046104, 14.74399146842491790500188504321, 15.804415325433289969930495126516, 16.54324751719986885188579635231, 17.15906880223015533506246378526, 17.72229207496142494659211223873, 18.43939566537931691965569831134, 19.54436323122246562734790592403, 20.34713464988536206952622907587, 21.18096538626483623698433681166

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