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Properties

Label 2-1205-5.4-c1-0-69
Degree $2$
Conductor $1205$
Sign $0.957 - 0.289i$
Analytic cond. $9.62197$
Root an. cond. $3.10193$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.779i·2-s + 2.96i·3-s + 1.39·4-s + (2.14 − 0.646i)5-s + 2.30·6-s − 0.937i·7-s − 2.64i·8-s − 5.77·9-s + (−0.503 − 1.66i)10-s + 1.14·11-s + 4.12i·12-s − 5.29i·13-s − 0.730·14-s + (1.91 + 6.34i)15-s + 0.724·16-s + 4.50i·17-s + ⋯
L(s)  = 1  − 0.551i·2-s + 1.71i·3-s + 0.696·4-s + (0.957 − 0.289i)5-s + 0.942·6-s − 0.354i·7-s − 0.934i·8-s − 1.92·9-s + (−0.159 − 0.527i)10-s + 0.346·11-s + 1.19i·12-s − 1.46i·13-s − 0.195·14-s + (0.494 + 1.63i)15-s + 0.181·16-s + 1.09i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.957 - 0.289i$
Analytic conductor: \(9.62197\)
Root analytic conductor: \(3.10193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1205} (724, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1205,\ (\ :1/2),\ 0.957 - 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.448964021\)
\(L(\frac12)\) \(\approx\) \(2.448964021\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.14 + 0.646i)T \)
241 \( 1 - T \)
good2 \( 1 + 0.779iT - 2T^{2} \)
3 \( 1 - 2.96iT - 3T^{2} \)
7 \( 1 + 0.937iT - 7T^{2} \)
11 \( 1 - 1.14T + 11T^{2} \)
13 \( 1 + 5.29iT - 13T^{2} \)
17 \( 1 - 4.50iT - 17T^{2} \)
19 \( 1 - 5.43T + 19T^{2} \)
23 \( 1 - 1.97iT - 23T^{2} \)
29 \( 1 - 7.61T + 29T^{2} \)
31 \( 1 + 9.12T + 31T^{2} \)
37 \( 1 - 5.51iT - 37T^{2} \)
41 \( 1 - 6.55T + 41T^{2} \)
43 \( 1 + 0.588iT - 43T^{2} \)
47 \( 1 + 2.18iT - 47T^{2} \)
53 \( 1 - 1.78iT - 53T^{2} \)
59 \( 1 + 12.8T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 13.0iT - 67T^{2} \)
71 \( 1 + 3.64T + 71T^{2} \)
73 \( 1 + 7.30iT - 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 14.6iT - 83T^{2} \)
89 \( 1 - 3.52T + 89T^{2} \)
97 \( 1 + 6.10iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.930509267364994700927884579396, −9.398694295381769014909634405303, −8.460032667279462616025238784764, −7.35690967477676762037426849040, −6.01801908965440360515252243370, −5.55620314205898053264436752076, −4.49379895206267985795255077287, −3.46667727346886042881787522668, −2.83363189186701633855091377957, −1.27900387139365055681688328317, 1.36160675106340002067474659970, 2.18690775965118710289746892289, 2.94454604360803464363636063411, 5.02068550586057563719540868126, 6.02628896985617272988812384927, 6.39033037299590191051241009155, 7.30386343488699664126252502403, 7.48455813249975762077279269873, 8.876811534975311808361429330671, 9.313745070790999721521430667789

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