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Properties

Label 1-645-645.617-r0-0-0
Degree $1$
Conductor $645$
Sign $0.601 + 0.798i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.601 + 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(645\)    =    \(3 \cdot 5 \cdot 43\)
Sign: $0.601 + 0.798i$
Analytic conductor: \(2.99536\)
Root analytic conductor: \(2.99536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{645} (617, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 645,\ (0:\ ),\ 0.601 + 0.798i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8583960738 + 0.4281501960i\)
\(L(\frac12)\) \(\approx\) \(0.8583960738 + 0.4281501960i\)
\(L(1)\) \(\approx\) \(0.7789153299 + 0.1422091183i\)
\(L(1)\) \(\approx\) \(0.7789153299 + 0.1422091183i\)

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

−22.447168159019413447559746501, −21.85713663794342250766616330343, −20.90655965711384493713732571601, −20.32245808412319977785871337520, −19.18444736839220531392394948498, −18.86192616002823677195589260630, −17.663971336003813237641921276082, −17.25695997871862627262949824809, −16.35232914442103467070952809669, −15.35253484228091289277937975018, −14.64287067750667378150909395563, −13.59763612658422174357391512014, −12.249416087576385559123686172951, −11.65203840136472127227167572272, −11.035284434327826856799850059341, −9.8609526352121242182144816048, −9.038787860608387418902504455191, −8.44837843452052203457756685782, −7.30388529871549291266498789962, −6.64111312990257119406283214475, −5.34275183857442756229838826895, −4.24895452512759619966827178979, −2.80465599736584211373404912612, −2.0468130095789201915835067699, −0.72928046907696686077822447720, 1.24062446610595607084687927908, 1.9693521946668929263206769613, 3.458673154711515814863197333176, 4.70290402559512880143673134425, 5.77636615132462479084086040567, 6.86168146486423814829799256339, 7.62591768834762903854651149130, 8.376196286036434773893746403224, 9.39660505590370024987050614557, 10.20153373445236689689324585443, 11.00641584414944294154601714312, 11.8642557786966821146892048092, 12.73802138879969575709938576296, 14.296932079522917879088713418021, 14.68018487811725576424598094848, 15.571451255261026324672154675532, 16.745349004091526941565208495488, 17.33463510931702967686109322987, 17.75448566539739365166237170118, 18.98780362192930103232893272465, 19.595651405613346485241047670754, 20.38867608836444387090825270786, 21.11215137772337598605877108666, 22.12422071715002935554865519570, 23.24180062249335278567513060960

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