Properties

Label 1-297-297.245-r1-0-0
Degree $1$
Conductor $297$
Sign $0.947 - 0.320i$
Analytic cond. $31.9170$
Root an. cond. $31.9170$
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  + (0.615 − 0.788i)2-s + (−0.241 − 0.970i)4-s + (−0.990 + 0.139i)5-s + (−0.997 + 0.0697i)7-s + (−0.913 − 0.406i)8-s + (−0.5 + 0.866i)10-s + (0.0348 + 0.999i)13-s + (−0.559 + 0.829i)14-s + (−0.882 + 0.469i)16-s + (0.978 + 0.207i)17-s + (0.913 + 0.406i)19-s + (0.374 + 0.927i)20-s + (−0.766 − 0.642i)23-s + (0.961 − 0.275i)25-s + (0.809 + 0.587i)26-s + ⋯
L(s)  = 1  + (0.615 − 0.788i)2-s + (−0.241 − 0.970i)4-s + (−0.990 + 0.139i)5-s + (−0.997 + 0.0697i)7-s + (−0.913 − 0.406i)8-s + (−0.5 + 0.866i)10-s + (0.0348 + 0.999i)13-s + (−0.559 + 0.829i)14-s + (−0.882 + 0.469i)16-s + (0.978 + 0.207i)17-s + (0.913 + 0.406i)19-s + (0.374 + 0.927i)20-s + (−0.766 − 0.642i)23-s + (0.961 − 0.275i)25-s + (0.809 + 0.587i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(31.9170\)
Root analytic conductor: \(31.9170\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 297,\ (1:\ ),\ 0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.480189997 - 0.2436569606i\)
\(L(\frac12)\) \(\approx\) \(1.480189997 - 0.2436569606i\)
\(L(1)\) \(\approx\) \(0.9948593557 - 0.3748682703i\)
\(L(1)\) \(\approx\) \(0.9948593557 - 0.3748682703i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.615 - 0.788i)T \)
5 \( 1 + (-0.990 + 0.139i)T \)
7 \( 1 + (-0.997 + 0.0697i)T \)
13 \( 1 + (0.0348 + 0.999i)T \)
17 \( 1 + (0.978 + 0.207i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.559 - 0.829i)T \)
31 \( 1 + (0.848 - 0.529i)T \)
37 \( 1 + (0.913 - 0.406i)T \)
41 \( 1 + (-0.559 + 0.829i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.241 - 0.970i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.719 - 0.694i)T \)
61 \( 1 + (0.848 + 0.529i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.978 + 0.207i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (-0.615 + 0.788i)T \)
83 \( 1 + (-0.0348 + 0.999i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.990 + 0.139i)T \)
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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

−25.2887418484310017610295364087, −24.14723332365544096240124459294, −23.47217471322352298464434957978, −22.60507779171733721752313263255, −22.119989969429836203205010081214, −20.694840632120214854119484463428, −19.936737285195150665420638176586, −18.86729657784457054574410387970, −17.80131460824859778568554820182, −16.654420755336367033309778388839, −15.91251670491536647977307226361, −15.37914000356294066684158447901, −14.242128607206636771221066737699, −13.20225952413567819623886804079, −12.389640199707939344505743473628, −11.62294176220077799967187709513, −10.12683395100115771401345683994, −8.92337147612162206266096240787, −7.80107810395485829178022107427, −7.17609852362334694859807928559, −5.926143262516950501634639505402, −4.96332439347763777722751757490, −3.60448082327049276898051337386, −3.08629864561581335334735856019, −0.510722817590360201617139162439, 0.87174957092658640807177979200, 2.53807521005900538200517440228, 3.60319694093377819224367358261, 4.321876248092087737467780946, 5.768372538162216695609976235387, 6.74672158322471242285599236600, 8.05822767509909024113250035162, 9.45119151724171186371813538361, 10.15191089185530808761907527255, 11.46510479132479270234915872369, 12.00981487457737984026245876943, 12.92323099884337492853800830906, 13.9952327597522052314848036949, 14.873190268105294883243736319135, 15.90025378911895627488677903976, 16.62713349745849159032205106648, 18.45596366910943865723069404391, 18.94134723047270674490053544093, 19.75279798930220518192996422926, 20.55211641428248275385468124043, 21.61962581107641774589293205466, 22.53843808484425143541546041869, 23.12920941658308415876669346585, 23.927944818888837777487221663617, 24.90087843511226484229451055079

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