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Properties

Label 1-43e2-1849.216-r0-0-0
Degree $1$
Conductor $1849$
Sign $-0.993 - 0.113i$
Analytic cond. $8.58671$
Root an. cond. $8.58671$
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  + (0.989 − 0.145i)2-s + (−0.934 − 0.357i)3-s + (0.957 − 0.288i)4-s + (−0.181 − 0.983i)5-s + (−0.976 − 0.217i)6-s + (−0.0365 − 0.999i)7-s + (0.905 − 0.424i)8-s + (0.744 + 0.667i)9-s + (−0.322 − 0.946i)10-s + (0.109 − 0.994i)11-s + (−0.997 − 0.0729i)12-s + (−0.976 + 0.217i)13-s + (−0.181 − 0.983i)14-s + (−0.181 + 0.983i)15-s + (0.833 − 0.551i)16-s + (0.520 − 0.853i)17-s + ⋯
L(s)  = 1  + (0.989 − 0.145i)2-s + (−0.934 − 0.357i)3-s + (0.957 − 0.288i)4-s + (−0.181 − 0.983i)5-s + (−0.976 − 0.217i)6-s + (−0.0365 − 0.999i)7-s + (0.905 − 0.424i)8-s + (0.744 + 0.667i)9-s + (−0.322 − 0.946i)10-s + (0.109 − 0.994i)11-s + (−0.997 − 0.0729i)12-s + (−0.976 + 0.217i)13-s + (−0.181 − 0.983i)14-s + (−0.181 + 0.983i)15-s + (0.833 − 0.551i)16-s + (0.520 − 0.853i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-0.993 - 0.113i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (216, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ -0.993 - 0.113i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1111895474 - 1.951357459i\)
\(L(\frac12)\) \(\approx\) \(0.1111895474 - 1.951357459i\)
\(L(1)\) \(\approx\) \(1.078350693 - 0.8998362106i\)
\(L(1)\) \(\approx\) \(1.078350693 - 0.8998362106i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.989 - 0.145i)T \)
3 \( 1 + (-0.934 - 0.357i)T \)
5 \( 1 + (-0.181 - 0.983i)T \)
7 \( 1 + (-0.0365 - 0.999i)T \)
11 \( 1 + (0.109 - 0.994i)T \)
13 \( 1 + (-0.976 + 0.217i)T \)
17 \( 1 + (0.520 - 0.853i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.997 + 0.0729i)T \)
29 \( 1 + (0.744 - 0.667i)T \)
31 \( 1 + (-0.976 - 0.217i)T \)
37 \( 1 + (-0.872 - 0.489i)T \)
41 \( 1 + (0.744 + 0.667i)T \)
47 \( 1 + (0.744 - 0.667i)T \)
53 \( 1 + (0.905 + 0.424i)T \)
59 \( 1 + (0.639 + 0.768i)T \)
61 \( 1 + (-0.0365 - 0.999i)T \)
67 \( 1 + (0.989 - 0.145i)T \)
71 \( 1 + (0.744 + 0.667i)T \)
73 \( 1 + (0.957 + 0.288i)T \)
79 \( 1 + (-0.791 + 0.611i)T \)
83 \( 1 + (-0.0365 - 0.999i)T \)
89 \( 1 + (-0.694 + 0.719i)T \)
97 \( 1 + (-0.976 - 0.217i)T \)
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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

−20.73404617517485216019254618297, −19.83790723835113387261682283474, −19.06095641899760229736531332866, −18.01657475308200123355369440030, −17.644799410535870693356832349120, −16.641695168362042067402515162264, −15.793491394865436955004971003877, −15.33763142112407263358537399108, −14.66908195005586396823658786453, −14.126051252659427441735050436685, −12.727452076319765158062237999105, −12.20043364448892043277761959876, −11.872062942475421078459886524168, −10.88544744728890149339494513994, −10.21436406557179457072840117941, −9.54958965465343642187899147476, −8.05965926294705735651615373706, −7.16768062843481174766891168294, −6.67753013150997680234302140769, −5.65367699219918376691683591164, −5.31456010148858897548688031586, −4.27195805266242021037473620605, −3.49118132384102249304359092469, −2.552338268555144262607469151968, −1.684688791558640159128715572469, 0.56780621390309782721261869971, 1.23561387155561150952433355972, 2.40072956859008274401652058005, 3.70361447798869517262570313087, 4.31117552439169179902703562937, 5.27066224424684248996489678874, 5.57787592130342678863520863278, 6.69751740004483123773669590925, 7.42000608752553020910501943647, 8.00995744950822635991698299274, 9.54513497783630259147827858698, 10.17977344590783046174210435735, 11.158205855031690894878706654768, 11.75715621183351520652323952519, 12.27957122250232322988192332917, 13.048411682902403578561063473001, 13.88500200045188582300610278535, 14.11492264599454290021453250123, 15.61442742637112952447126481819, 16.22862832959739916757302772650, 16.651073070375362701358832394669, 17.266604061662357759280888623889, 18.31960402930233725754791614031, 19.33445155543282855267849051621, 19.864296267374430005868009483724

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