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Properties

Label 1-245-245.152-r0-0-0
Degree $1$
Conductor $245$
Sign $0.738 - 0.673i$
Analytic cond. $1.13777$
Root an. cond. $1.13777$
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  + (0.294 + 0.955i)2-s + (−0.930 + 0.365i)3-s + (−0.826 + 0.563i)4-s + (−0.623 − 0.781i)6-s + (−0.781 − 0.623i)8-s + (0.733 − 0.680i)9-s + (−0.733 − 0.680i)11-s + (0.563 − 0.826i)12-s + (−0.974 − 0.222i)13-s + (0.365 − 0.930i)16-s + (−0.997 − 0.0747i)17-s + (0.866 + 0.5i)18-s + (−0.5 − 0.866i)19-s + (0.433 − 0.900i)22-s + (0.997 − 0.0747i)23-s + (0.955 + 0.294i)24-s + ⋯
L(s)  = 1  + (0.294 + 0.955i)2-s + (−0.930 + 0.365i)3-s + (−0.826 + 0.563i)4-s + (−0.623 − 0.781i)6-s + (−0.781 − 0.623i)8-s + (0.733 − 0.680i)9-s + (−0.733 − 0.680i)11-s + (0.563 − 0.826i)12-s + (−0.974 − 0.222i)13-s + (0.365 − 0.930i)16-s + (−0.997 − 0.0747i)17-s + (0.866 + 0.5i)18-s + (−0.5 − 0.866i)19-s + (0.433 − 0.900i)22-s + (0.997 − 0.0747i)23-s + (0.955 + 0.294i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.738 - 0.673i$
Analytic conductor: \(1.13777\)
Root analytic conductor: \(1.13777\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 245,\ (0:\ ),\ 0.738 - 0.673i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3566845444 - 0.1382204851i\)
\(L(\frac12)\) \(\approx\) \(0.3566845444 - 0.1382204851i\)
\(L(1)\) \(\approx\) \(0.5820898119 + 0.2394873469i\)
\(L(1)\) \(\approx\) \(0.5820898119 + 0.2394873469i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.294 + 0.955i)T \)
3 \( 1 + (-0.930 + 0.365i)T \)
11 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (-0.974 - 0.222i)T \)
17 \( 1 + (-0.997 - 0.0747i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.997 - 0.0747i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.563 + 0.826i)T \)
41 \( 1 + (-0.623 + 0.781i)T \)
43 \( 1 + (0.781 - 0.623i)T \)
47 \( 1 + (-0.294 - 0.955i)T \)
53 \( 1 + (-0.563 - 0.826i)T \)
59 \( 1 + (-0.988 + 0.149i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.900 - 0.433i)T \)
73 \( 1 + (-0.294 + 0.955i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.974 - 0.222i)T \)
89 \( 1 + (-0.733 + 0.680i)T \)
97 \( 1 - iT \)
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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

−26.60958458045617218049756745454, −25.05512869987425605834869880565, −24.053382578839137783449384770509, −23.2321023902079394658663207044, −22.579096633468052461270327647213, −21.64312381488789538638561847671, −20.85497565929935230807707251749, −19.619762982641418814921753058119, −18.91335572489424077114402539070, −17.82844412672512158599526431040, −17.30925620589986894250542934595, −15.89118693399251640313478947801, −14.75379147369837912438790755823, −13.56486104547646528628548921369, −12.558351313703061659052572887373, −12.12605003362151380069768663085, −10.82328096044015685092651116792, −10.31241803264275178958328708024, −9.04775003255072275755990636816, −7.55446009836532284462130141085, −6.34575460017368103486882320401, −5.0775113756164521640985387876, −4.44330031389897791271690755663, −2.677187201649536542406945878305, −1.55439963313996382305728230971, 0.27689382865914029010634705372, 2.92664773912408678834153049963, 4.49765285212278588517036359509, 5.10308496388493942315445814127, 6.262769043247283838018571925396, 7.08518794283687644403157510973, 8.362752949304806948525093731154, 9.49718688107224666718801185613, 10.63054787725788837621438656925, 11.74792488779973245573496877236, 12.85283148955669815382264314594, 13.63905290290268370917911380564, 15.16397362896791281988831861634, 15.5308627286066106055337181879, 16.70325637080233025965548607898, 17.32055386467740119395999054370, 18.17237481101802674727187589127, 19.25517124300747069089815228777, 20.88870692034594393206600160066, 21.76081483631574408718681636855, 22.38409258889401782872302732914, 23.34946318760504163940349735778, 24.10163315279427929113749144029, 24.8043638020620228064443585694, 26.18079365026837540645514177

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