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Properties

Label 2-1205-5.4-c1-0-110
Degree $2$
Conductor $1205$
Sign $0.247 - 0.968i$
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  − 2.44i·2-s − 0.992i·3-s − 3.99·4-s + (0.553 − 2.16i)5-s − 2.42·6-s − 0.744i·7-s + 4.88i·8-s + 2.01·9-s + (−5.30 − 1.35i)10-s − 2.60·11-s + 3.96i·12-s − 4.18i·13-s − 1.82·14-s + (−2.14 − 0.549i)15-s + 3.96·16-s + 0.484i·17-s + ⋯
L(s)  = 1  − 1.73i·2-s − 0.572i·3-s − 1.99·4-s + (0.247 − 0.968i)5-s − 0.991·6-s − 0.281i·7-s + 1.72i·8-s + 0.671·9-s + (−1.67 − 0.428i)10-s − 0.785·11-s + 1.14i·12-s − 1.16i·13-s − 0.487·14-s + (−0.555 − 0.141i)15-s + 0.991·16-s + 0.117i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.247 - 0.968i$
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: \((2,\ 1205,\ (\ :1/2),\ 0.247 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.107741247\)
\(L(\frac12)\) \(\approx\) \(1.107741247\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.553 + 2.16i)T \)
241 \( 1 - T \)
good2 \( 1 + 2.44iT - 2T^{2} \)
3 \( 1 + 0.992iT - 3T^{2} \)
7 \( 1 + 0.744iT - 7T^{2} \)
11 \( 1 + 2.60T + 11T^{2} \)
13 \( 1 + 4.18iT - 13T^{2} \)
17 \( 1 - 0.484iT - 17T^{2} \)
19 \( 1 + 3.30T + 19T^{2} \)
23 \( 1 - 1.44iT - 23T^{2} \)
29 \( 1 - 0.129T + 29T^{2} \)
31 \( 1 - 4.43T + 31T^{2} \)
37 \( 1 - 4.67iT - 37T^{2} \)
41 \( 1 + 9.21T + 41T^{2} \)
43 \( 1 - 7.65iT - 43T^{2} \)
47 \( 1 + 2.28iT - 47T^{2} \)
53 \( 1 + 12.5iT - 53T^{2} \)
59 \( 1 - 7.30T + 59T^{2} \)
61 \( 1 + 5.28T + 61T^{2} \)
67 \( 1 - 5.69iT - 67T^{2} \)
71 \( 1 - 2.08T + 71T^{2} \)
73 \( 1 + 15.5iT - 73T^{2} \)
79 \( 1 - 10.0T + 79T^{2} \)
83 \( 1 - 2.71iT - 83T^{2} \)
89 \( 1 - 9.86T + 89T^{2} \)
97 \( 1 - 6.26iT - 97T^{2} \)
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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

−9.401380956731068665724464670537, −8.360491254456545298925369777898, −7.902576813813983160956369384439, −6.54681789244798459937644881938, −5.26878901000184462141300123868, −4.59688031504142450488867245131, −3.57705477650551374675894661834, −2.43196460682579917984319599592, −1.47615876386450421813123870157, −0.48868793715585744736763339993, 2.29650765086900775099791787661, 3.82509110838840675181043107037, 4.62976051021671599236528961803, 5.50419019875471041347085932266, 6.41359517511781762585030267690, 6.97944506881720810436098154073, 7.67917423009620877708844341163, 8.678758319065396506600493601848, 9.342815586523772291723853187034, 10.19138681019579048887114323215

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