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Properties

Label 1-645-645.284-r0-0-0
Degree $1$
Conductor $645$
Sign $-0.930 - 0.365i$
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  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.5 − 0.866i)7-s + (0.222 − 0.974i)8-s + (−0.623 − 0.781i)11-s + (−0.955 + 0.294i)13-s + (−0.826 − 0.563i)14-s + (−0.222 − 0.974i)16-s + (−0.733 + 0.680i)17-s + (−0.365 − 0.930i)19-s + (−0.900 − 0.433i)22-s + (−0.988 + 0.149i)23-s + (−0.733 + 0.680i)26-s + (−0.988 − 0.149i)28-s + (0.826 + 0.563i)29-s + ⋯
L(s)  = 1  + (0.900 − 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.5 − 0.866i)7-s + (0.222 − 0.974i)8-s + (−0.623 − 0.781i)11-s + (−0.955 + 0.294i)13-s + (−0.826 − 0.563i)14-s + (−0.222 − 0.974i)16-s + (−0.733 + 0.680i)17-s + (−0.365 − 0.930i)19-s + (−0.900 − 0.433i)22-s + (−0.988 + 0.149i)23-s + (−0.733 + 0.680i)26-s + (−0.988 − 0.149i)28-s + (0.826 + 0.563i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 645 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.930 - 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(645\)    =    \(3 \cdot 5 \cdot 43\)
Sign: $-0.930 - 0.365i$
Analytic conductor: \(2.99536\)
Root analytic conductor: \(2.99536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{645} (284, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 645,\ (0:\ ),\ -0.930 - 0.365i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2790518770 - 1.476154809i\)
\(L(\frac12)\) \(\approx\) \(0.2790518770 - 1.476154809i\)
\(L(1)\) \(\approx\) \(1.129867037 - 0.7693790556i\)
\(L(1)\) \(\approx\) \(1.129867037 - 0.7693790556i\)

Euler product

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

−23.01770080010682167414881086488, −22.58817148956087612618176236638, −21.74480885797195168299397839557, −21.01704586725246913913457608715, −20.10905935554953033461674847517, −19.31744823716611422374720935005, −18.05278527999699931896601442657, −17.50486124019296481049093789672, −16.212508826406803121247335736179, −15.78411591975209768527462930659, −14.91853963776488346850638540861, −14.19862719081881815467153144627, −13.13331513282184264592186363907, −12.344111847977625714223643083362, −11.96527981527648596828087745992, −10.60851770930636564116481107164, −9.67883817314559816216778286470, −8.51419829456729024901308819583, −7.60609179352661850173759778940, −6.72060918023831508745687541556, −5.77517850274689164642732411450, −4.9778242902816851440682368935, −4.046448069687928080140181292050, −2.70827797523735558601035712128, −2.20504575150280444052707501023, 0.48964439870852115195712151959, 2.054721976691046121776531931559, 2.98859286457993790264687822127, 4.03499800494324749616830051253, 4.78356604203633373375442044261, 5.94834701741737167939276272782, 6.74635166580226377963249032926, 7.660097065265769108191082201298, 9.00621301432549301955287927374, 10.16639787854634442434100262597, 10.65810790630850958285538710455, 11.627919674298703590831383292584, 12.56386863884867465583161249851, 13.424199632231779352263427854502, 13.86439480094660024725592486252, 14.950438833665794494654087559237, 15.747600707618046164341636134950, 16.56817302423198355193945371601, 17.48235929512836328441954596077, 18.71419949162979656036139598860, 19.61556042393186165817162680800, 19.94336800520960082441052762144, 21.024109449160432653692926889779, 21.84415825706667147835173825370, 22.32382783647167831850957902496

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