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Properties

Label 1-1205-1205.1042-r0-0-0
Degree $1$
Conductor $1205$
Sign $0.267 - 0.963i$
Analytic cond. $5.59599$
Root an. cond. $5.59599$
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)2-s + (−0.998 + 0.0523i)3-s + (0.866 − 0.5i)4-s + (−0.951 + 0.309i)6-s + (0.999 − 0.0261i)7-s + (0.707 − 0.707i)8-s + (0.994 − 0.104i)9-s + (−0.608 − 0.793i)11-s + (−0.838 + 0.544i)12-s + (0.333 − 0.942i)13-s + (0.958 − 0.284i)14-s + (0.5 − 0.866i)16-s + (0.972 + 0.233i)17-s + (0.933 − 0.358i)18-s + (0.793 − 0.608i)19-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.998 + 0.0523i)3-s + (0.866 − 0.5i)4-s + (−0.951 + 0.309i)6-s + (0.999 − 0.0261i)7-s + (0.707 − 0.707i)8-s + (0.994 − 0.104i)9-s + (−0.608 − 0.793i)11-s + (−0.838 + 0.544i)12-s + (0.333 − 0.942i)13-s + (0.958 − 0.284i)14-s + (0.5 − 0.866i)16-s + (0.972 + 0.233i)17-s + (0.933 − 0.358i)18-s + (0.793 − 0.608i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.267 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.267 - 0.963i$
Analytic conductor: \(5.59599\)
Root analytic conductor: \(5.59599\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (1042, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1205,\ (0:\ ),\ 0.267 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.948660488 - 1.481256526i\)
\(L(\frac12)\) \(\approx\) \(1.948660488 - 1.481256526i\)
\(L(1)\) \(\approx\) \(1.553159060 - 0.5417125950i\)
\(L(1)\) \(\approx\) \(1.553159060 - 0.5417125950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (-0.998 + 0.0523i)T \)
7 \( 1 + (0.999 - 0.0261i)T \)
11 \( 1 + (-0.608 - 0.793i)T \)
13 \( 1 + (0.333 - 0.942i)T \)
17 \( 1 + (0.972 + 0.233i)T \)
19 \( 1 + (0.793 - 0.608i)T \)
23 \( 1 + (-0.0784 + 0.996i)T \)
29 \( 1 + (-0.629 + 0.777i)T \)
31 \( 1 + (-0.725 - 0.688i)T \)
37 \( 1 + (-0.333 - 0.942i)T \)
41 \( 1 + (-0.987 + 0.156i)T \)
43 \( 1 + (0.852 + 0.522i)T \)
47 \( 1 + (-0.156 + 0.987i)T \)
53 \( 1 + (-0.358 - 0.933i)T \)
59 \( 1 + (-0.998 - 0.0523i)T \)
61 \( 1 + (-0.453 + 0.891i)T \)
67 \( 1 + (0.933 + 0.358i)T \)
71 \( 1 + (0.878 - 0.477i)T \)
73 \( 1 + (0.649 - 0.760i)T \)
79 \( 1 + (-0.453 - 0.891i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.991 - 0.130i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
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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

−21.43381679835829703808763231985, −20.77993217086481552291948396833, −20.27558835115471662703175915708, −18.63464655368119271265692004599, −18.36751518835480399009536181636, −17.13959633350086054264349720520, −16.79026004269275128478503281998, −15.8611266068469900730058720606, −15.20864994337013161289492072677, −14.26003960435771242333368377249, −13.69304885403063397899557747933, −12.497524506296549977908877650925, −12.12956993041147184312085942895, −11.353171764396732561008317937943, −10.64593827859203641447682242060, −9.76462779332479246011495392937, −8.28081159404547594070163058903, −7.47634734635886947364181465725, −6.83851845533382719154442477185, −5.7975101813261763126200382839, −5.13337401033182752710510347155, −4.52513966322155276244847589809, −3.60274864480649996872761914674, −2.14907336219405794148068627052, −1.40773362493734109954897105997, 0.87081267709877372371480776623, 1.72543704220079576417172059326, 3.10997278571980063096656307882, 3.88128890358105950883758031530, 5.173366542528995428924252019174, 5.34746114822760400250851037905, 6.1295825932873777118943821773, 7.42225972064208280953135242880, 7.85244645870087467291639979042, 9.41458446443149912452869644048, 10.4776321365019179473985033783, 11.02258308053368158320385622196, 11.526242291079283648807924797328, 12.4440549775997283293708257211, 13.137523644233730243420508161205, 13.91484435178981090651262061906, 14.83361899552515749456009894677, 15.60937576808415425784104331904, 16.22590865703723591950892683949, 17.092364116203313606547014270529, 18.01297904789285200516303156288, 18.58987858774127999334521096619, 19.62502245870813287725647689345, 20.62959346425103394699776696423, 21.143920828159368319255297136110

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