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Properties

Label 1-547-547.34-r0-0-0
Degree $1$
Conductor $547$
Sign $-0.732 - 0.680i$
Analytic cond. $2.54025$
Root an. cond. $2.54025$
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  + (−0.177 + 0.984i)2-s + (−0.900 + 0.433i)3-s + (−0.937 − 0.349i)4-s + (0.490 − 0.871i)5-s + (−0.267 − 0.963i)6-s + (−0.459 − 0.888i)7-s + (0.509 − 0.860i)8-s + (0.623 − 0.781i)9-s + (0.770 + 0.636i)10-s + (−0.996 + 0.0804i)11-s + (0.995 − 0.0919i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (−0.0632 + 0.997i)15-s + (0.756 + 0.654i)16-s + (0.973 + 0.228i)17-s + ⋯
L(s)  = 1  + (−0.177 + 0.984i)2-s + (−0.900 + 0.433i)3-s + (−0.937 − 0.349i)4-s + (0.490 − 0.871i)5-s + (−0.267 − 0.963i)6-s + (−0.459 − 0.888i)7-s + (0.509 − 0.860i)8-s + (0.623 − 0.781i)9-s + (0.770 + 0.636i)10-s + (−0.996 + 0.0804i)11-s + (0.995 − 0.0919i)12-s + (−0.733 − 0.680i)13-s + (0.955 − 0.294i)14-s + (−0.0632 + 0.997i)15-s + (0.756 + 0.654i)16-s + (0.973 + 0.228i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(547\)
Sign: $-0.732 - 0.680i$
Analytic conductor: \(2.54025\)
Root analytic conductor: \(2.54025\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{547} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 547,\ (0:\ ),\ -0.732 - 0.680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.04323225234 - 0.1101334364i\)
\(L(\frac12)\) \(\approx\) \(0.04323225234 - 0.1101334364i\)
\(L(1)\) \(\approx\) \(0.5141721046 + 0.1127681883i\)
\(L(1)\) \(\approx\) \(0.5141721046 + 0.1127681883i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad547 \( 1 \)
good2 \( 1 + (-0.177 + 0.984i)T \)
3 \( 1 + (-0.900 + 0.433i)T \)
5 \( 1 + (0.490 - 0.871i)T \)
7 \( 1 + (-0.459 - 0.888i)T \)
11 \( 1 + (-0.996 + 0.0804i)T \)
13 \( 1 + (-0.733 - 0.680i)T \)
17 \( 1 + (0.973 + 0.228i)T \)
19 \( 1 + (0.983 + 0.183i)T \)
23 \( 1 + (-0.910 - 0.413i)T \)
29 \( 1 + (-0.985 - 0.171i)T \)
31 \( 1 + (-0.596 + 0.802i)T \)
37 \( 1 + (-0.890 - 0.454i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.667 + 0.744i)T \)
47 \( 1 + (0.948 + 0.316i)T \)
53 \( 1 + (0.895 - 0.444i)T \)
59 \( 1 + (-0.0402 + 0.999i)T \)
61 \( 1 + (-0.880 + 0.474i)T \)
67 \( 1 + (-0.577 - 0.816i)T \)
71 \( 1 + (-0.109 + 0.994i)T \)
73 \( 1 + (-0.0632 - 0.997i)T \)
79 \( 1 + (0.978 + 0.205i)T \)
83 \( 1 + (-0.632 - 0.774i)T \)
89 \( 1 + (0.851 - 0.524i)T \)
97 \( 1 + (0.300 - 0.953i)T \)
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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

−23.45983498788422508724224632540, −22.51988302258520582570093866784, −22.01709133069641060290885246330, −21.50596782086980338449005879041, −20.4124504932328581255489481478, −19.06683463522884290707321546271, −18.674908017773206601222609835800, −18.18636148139963679729878699936, −17.24758693839484156829976829461, −16.35539496304831877110055255420, −15.20736045865267792913641906041, −13.945466522948557367533253746942, −13.33007499302550794616529468440, −12.195980525957349036306649162628, −11.82162950353466295716562017668, −10.78294751073781262067432995805, −9.996034697835372077941716545691, −9.35409596552177097168376756971, −7.83314230598148868669013352400, −7.04053413940899907081001770788, −5.59930571615650628466463389398, −5.29697452251653856970273792100, −3.601017633781641364041777013823, −2.509388692994208484357094286796, −1.79408865576526511260848613704, 0.079503641647049988113033142917, 1.26163412778905380683949723822, 3.4988777703520602716765546951, 4.63148920064805746882752378557, 5.37821084387726689484754745505, 5.955589104916689029800297711243, 7.23435425587920399373480132411, 7.91463953732963766449458985700, 9.26128563311328114364856034079, 10.11541357234519711976181761541, 10.39784210938288691384599431611, 12.16103949430624651377797381483, 12.86642885718537015397250775548, 13.66361529310890147798559856024, 14.74513784410908555282952299077, 15.811389759493317237386641460591, 16.4503839966891608761833089179, 16.89171817011151347120430506408, 17.80297765103976806779982108495, 18.36742544873160663040691719929, 19.76860893301579365518604137207, 20.634863347365256838241448652739, 21.585325128016931588947574842490, 22.50593662491013374124408275911, 23.15055050799140435949793446575

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