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Properties

Label 2-1205-5.4-c1-0-57
Degree $2$
Conductor $1205$
Sign $0.991 - 0.132i$
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  − 1.30i·2-s + 1.88i·3-s + 0.287·4-s + (2.21 − 0.296i)5-s + 2.46·6-s + 1.28i·7-s − 2.99i·8-s − 0.550·9-s + (−0.387 − 2.90i)10-s + 0.698·11-s + 0.541i·12-s + 5.53i·13-s + 1.68·14-s + (0.558 + 4.17i)15-s − 3.34·16-s + 3.43i·17-s + ⋯
L(s)  = 1  − 0.925i·2-s + 1.08i·3-s + 0.143·4-s + (0.991 − 0.132i)5-s + 1.00·6-s + 0.486i·7-s − 1.05i·8-s − 0.183·9-s + (−0.122 − 0.917i)10-s + 0.210·11-s + 0.156i·12-s + 1.53i·13-s + 0.450·14-s + (0.144 + 1.07i)15-s − 0.835·16-s + 0.832i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.132i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.991 - 0.132i$
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: \((2,\ 1205,\ (\ :1/2),\ 0.991 - 0.132i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.358333739\)
\(L(\frac12)\) \(\approx\) \(2.358333739\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.21 + 0.296i)T \)
241 \( 1 - T \)
good2 \( 1 + 1.30iT - 2T^{2} \)
3 \( 1 - 1.88iT - 3T^{2} \)
7 \( 1 - 1.28iT - 7T^{2} \)
11 \( 1 - 0.698T + 11T^{2} \)
13 \( 1 - 5.53iT - 13T^{2} \)
17 \( 1 - 3.43iT - 17T^{2} \)
19 \( 1 - 3.98T + 19T^{2} \)
23 \( 1 + 6.27iT - 23T^{2} \)
29 \( 1 + 6.15T + 29T^{2} \)
31 \( 1 + 0.212T + 31T^{2} \)
37 \( 1 + 5.91iT - 37T^{2} \)
41 \( 1 + 9.65T + 41T^{2} \)
43 \( 1 - 7.78iT - 43T^{2} \)
47 \( 1 - 3.86iT - 47T^{2} \)
53 \( 1 + 10.2iT - 53T^{2} \)
59 \( 1 - 9.54T + 59T^{2} \)
61 \( 1 - 8.82T + 61T^{2} \)
67 \( 1 - 7.38iT - 67T^{2} \)
71 \( 1 + 6.00T + 71T^{2} \)
73 \( 1 - 2.96iT - 73T^{2} \)
79 \( 1 - 6.80T + 79T^{2} \)
83 \( 1 + 16.1iT - 83T^{2} \)
89 \( 1 + 1.74T + 89T^{2} \)
97 \( 1 - 0.714iT - 97T^{2} \)
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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

−9.868993557412429223647755827549, −9.289183044109370363108013018066, −8.677508049544373094612123957493, −7.07914812237141728835897623482, −6.35228252463306367568935505357, −5.35470884692701745471667611613, −4.35358721055342903067189458300, −3.58719160903292819778199852450, −2.37098855984390818616435634682, −1.54995403263144312936124638866, 1.12163881423772169592995773171, 2.22392757947007852824633921983, 3.33296330789463447162994436881, 5.24148415304930476212856095983, 5.60314491096921305429129312047, 6.59692105662212313887846420777, 7.26866732615519515054083238266, 7.64312606185700420666495583141, 8.620419793340989438818882231147, 9.706350743085815678316293706855

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