Properties

Label 1-837-837.148-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.210 + 0.977i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (0.559 − 0.829i)2-s + (−0.374 − 0.927i)4-s + (0.766 − 0.642i)5-s + (0.0348 + 0.999i)7-s + (−0.978 − 0.207i)8-s + (−0.104 − 0.994i)10-s + (−0.0348 − 0.999i)11-s + (−0.438 + 0.898i)13-s + (0.848 + 0.529i)14-s + (−0.719 + 0.694i)16-s + (−0.309 − 0.951i)17-s + (0.913 + 0.406i)19-s + (−0.882 − 0.469i)20-s + (−0.848 − 0.529i)22-s + (−0.848 − 0.529i)23-s + ⋯
L(s)  = 1  + (0.559 − 0.829i)2-s + (−0.374 − 0.927i)4-s + (0.766 − 0.642i)5-s + (0.0348 + 0.999i)7-s + (−0.978 − 0.207i)8-s + (−0.104 − 0.994i)10-s + (−0.0348 − 0.999i)11-s + (−0.438 + 0.898i)13-s + (0.848 + 0.529i)14-s + (−0.719 + 0.694i)16-s + (−0.309 − 0.951i)17-s + (0.913 + 0.406i)19-s + (−0.882 − 0.469i)20-s + (−0.848 − 0.529i)22-s + (−0.848 − 0.529i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.210 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.210 + 0.977i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.210 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2189088572 - 0.2710740594i\)
\(L(\frac12)\) \(\approx\) \(-0.2189088572 - 0.2710740594i\)
\(L(1)\) \(\approx\) \(0.9876518619 - 0.6603654832i\)
\(L(1)\) \(\approx\) \(0.9876518619 - 0.6603654832i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.559 - 0.829i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.0348 + 0.999i)T \)
11 \( 1 + (-0.0348 - 0.999i)T \)
13 \( 1 + (-0.438 + 0.898i)T \)
17 \( 1 + (-0.309 - 0.951i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (-0.848 - 0.529i)T \)
29 \( 1 + (-0.559 + 0.829i)T \)
37 \( 1 - T \)
41 \( 1 + (-0.719 - 0.694i)T \)
43 \( 1 + (0.997 + 0.0697i)T \)
47 \( 1 + (-0.241 - 0.970i)T \)
53 \( 1 + (-0.669 + 0.743i)T \)
59 \( 1 + (0.559 + 0.829i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.990 + 0.139i)T \)
83 \( 1 + (0.997 + 0.0697i)T \)
89 \( 1 + (-0.669 - 0.743i)T \)
97 \( 1 + (-0.882 - 0.469i)T \)
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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

−22.41734226087075471925213226896, −22.15916613708944357296001433550, −20.98317188732136687272773639320, −20.367551270234080388272063636748, −19.367283616030337531041323471072, −18.0112409380537078043423632132, −17.56481765461229991590563897638, −17.12199356533700346906653173308, −15.94083013745665737606157274815, −15.12988712355919292595918624232, −14.49851198744121398832682336618, −13.64702979740360806280567209526, −13.111708268092114893118629368916, −12.188901212816627758690988189333, −11.017069614096442268157375812444, −10.05456127408046932955043803290, −9.45131950808934312807357689961, −7.98706228975721332695696766648, −7.40965973575369403806203231358, −6.62385845355253251599439910123, −5.74419539635118372904505381336, −4.84155565855498091280782898200, −3.855617594754853829625394706347, −2.92812274093600477613121466680, −1.71424550271498886108095904047, 0.05659150596336323812824440197, 1.37877005095822635861955348465, 2.23637875577299305712192896262, 3.102601636767707101263147994082, 4.33546043394269795672471839910, 5.34811954606194545458972158444, 5.72643628418813415680411037975, 6.83793856838940239722886129640, 8.49862263896423283908987885136, 9.12805083613168482384610516059, 9.759485699519633030264004444773, 10.80835500081141453206578762830, 11.88462259943222256115802766049, 12.16911989203953275482275254265, 13.27275913565212192797555128101, 13.9765652256103845467590059852, 14.5004717276425015273121242345, 15.82074353883579675476071575369, 16.358564776790773701650819857003, 17.58379798210585683107017027338, 18.491319141332996609367599326031, 18.87609287921491284388322916128, 20.03053041358693164969800227652, 20.67935683629289530099050142518, 21.453475091924973937519511644272

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