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Properties

Label 1-645-645.152-r0-0-0
Degree $1$
Conductor $645$
Sign $0.983 + 0.180i$
Analytic cond. $2.99536$
Root an. cond. $2.99536$
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  + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.866 − 0.5i)7-s + (0.781 + 0.623i)8-s + (0.900 − 0.433i)11-s + (0.930 − 0.365i)13-s + (−0.733 − 0.680i)14-s + (0.623 + 0.781i)16-s + (0.149 + 0.988i)17-s + (−0.0747 − 0.997i)19-s + (0.974 − 0.222i)22-s + (−0.563 + 0.826i)23-s + (0.988 − 0.149i)26-s + (−0.563 − 0.826i)28-s + (−0.733 − 0.680i)29-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s + (0.900 + 0.433i)4-s + (−0.866 − 0.5i)7-s + (0.781 + 0.623i)8-s + (0.900 − 0.433i)11-s + (0.930 − 0.365i)13-s + (−0.733 − 0.680i)14-s + (0.623 + 0.781i)16-s + (0.149 + 0.988i)17-s + (−0.0747 − 0.997i)19-s + (0.974 − 0.222i)22-s + (−0.563 + 0.826i)23-s + (0.988 − 0.149i)26-s + (−0.563 − 0.826i)28-s + (−0.733 − 0.680i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(645\)    =    \(3 \cdot 5 \cdot 43\)
Sign: $0.983 + 0.180i$
Analytic conductor: \(2.99536\)
Root analytic conductor: \(2.99536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{645} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 645,\ (0:\ ),\ 0.983 + 0.180i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.680392542 + 0.2443312049i\)
\(L(\frac12)\) \(\approx\) \(2.680392542 + 0.2443312049i\)
\(L(1)\) \(\approx\) \(1.912630826 + 0.1722194105i\)
\(L(1)\) \(\approx\) \(1.912630826 + 0.1722194105i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.974 + 0.222i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.900 - 0.433i)T \)
13 \( 1 + (0.930 - 0.365i)T \)
17 \( 1 + (0.149 + 0.988i)T \)
19 \( 1 + (-0.0747 - 0.997i)T \)
23 \( 1 + (-0.563 + 0.826i)T \)
29 \( 1 + (-0.733 - 0.680i)T \)
31 \( 1 + (0.955 - 0.294i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.222 - 0.974i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (0.930 + 0.365i)T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (0.955 + 0.294i)T \)
67 \( 1 + (-0.997 + 0.0747i)T \)
71 \( 1 + (-0.826 + 0.563i)T \)
73 \( 1 + (0.930 - 0.365i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + (-0.680 - 0.733i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 + (-0.433 - 0.900i)T \)
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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

−22.73698765394224074265264252433, −22.249393528860711806133815517831, −21.29536917317512817341007546264, −20.49027944054569933474138543513, −19.79393179506831205850083277590, −18.86730799122712073367133936091, −18.1878206798735970345836255601, −16.514183348110518733698345589650, −16.330889501224217605556175691012, −15.22231028358628553247219399874, −14.4557943913110654739562698089, −13.64956572304510282653555725643, −12.7713726751742648400644885957, −12.01283307843420794545274381537, −11.37317500191000216547254069997, −10.147815451140595927689724209273, −9.46676214090591212578361875109, −8.271873134632037902665168504743, −6.86842283106062266452042690435, −6.36636116842531640404426783897, −5.429937387245119547879686236673, −4.25287997610039768078478906019, −3.4861195241146618004207190008, −2.460993481541867988748117635513, −1.299192714889921677424286919046, 1.20224090874876787148479069838, 2.644799311383625854669019484831, 3.74006327144349370404744970081, 4.14638109340129241435131785436, 5.73578082979984739177010544693, 6.21674737345025122434876883871, 7.14779704223764692363777573900, 8.15171413516232743906624143876, 9.25418013570993160103159065799, 10.42150317750901565192947320706, 11.24318380424351143551288684267, 12.080738718451562455361347103, 13.23785238524211425971731556114, 13.443748081021095677437211789449, 14.52940107580294259219495153378, 15.43540975135633749229377993880, 16.11677171758379103825329078981, 16.94806757637177458724632451599, 17.66777732712371329279927006002, 19.22856554970116472024665972590, 19.63629940208075666365303740351, 20.57928138993287842322798505220, 21.447226638862992261646791056771, 22.24146250136605357036613353089, 22.83359282057446947265166165754

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