Properties

Label 2-1205-5.4-c1-0-60
Degree $2$
Conductor $1205$
Sign $0.742 - 0.669i$
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  + 1.84i·2-s − 1.69i·3-s − 1.39·4-s + (1.66 − 1.49i)5-s + 3.12·6-s − 1.54i·7-s + 1.11i·8-s + 0.120·9-s + (2.75 + 3.05i)10-s + 2.53·11-s + 2.36i·12-s + 6.36i·13-s + 2.83·14-s + (−2.54 − 2.81i)15-s − 4.84·16-s + 3.54i·17-s + ⋯
L(s)  = 1  + 1.30i·2-s − 0.979i·3-s − 0.696·4-s + (0.742 − 0.669i)5-s + 1.27·6-s − 0.582i·7-s + 0.395i·8-s + 0.0402·9-s + (0.872 + 0.967i)10-s + 0.764·11-s + 0.682i·12-s + 1.76i·13-s + 0.758·14-s + (−0.656 − 0.727i)15-s − 1.21·16-s + 0.859i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.742 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1205\)    =    \(5 \cdot 241\)
Sign: $0.742 - 0.669i$
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: \((2,\ 1205,\ (\ :1/2),\ 0.742 - 0.669i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.137138805\)
\(L(\frac12)\) \(\approx\) \(2.137138805\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.66 + 1.49i)T \)
241 \( 1 - T \)
good2 \( 1 - 1.84iT - 2T^{2} \)
3 \( 1 + 1.69iT - 3T^{2} \)
7 \( 1 + 1.54iT - 7T^{2} \)
11 \( 1 - 2.53T + 11T^{2} \)
13 \( 1 - 6.36iT - 13T^{2} \)
17 \( 1 - 3.54iT - 17T^{2} \)
19 \( 1 + 0.713T + 19T^{2} \)
23 \( 1 - 1.66iT - 23T^{2} \)
29 \( 1 - 8.23T + 29T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 - 0.496iT - 37T^{2} \)
41 \( 1 + 8.92T + 41T^{2} \)
43 \( 1 - 1.50iT - 43T^{2} \)
47 \( 1 + 6.94iT - 47T^{2} \)
53 \( 1 + 9.87iT - 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 + 9.13T + 61T^{2} \)
67 \( 1 - 1.85iT - 67T^{2} \)
71 \( 1 + 9.47T + 71T^{2} \)
73 \( 1 + 8.88iT - 73T^{2} \)
79 \( 1 - 6.12T + 79T^{2} \)
83 \( 1 + 0.835iT - 83T^{2} \)
89 \( 1 + 2.27T + 89T^{2} \)
97 \( 1 - 7.56iT - 97T^{2} \)
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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

−9.578065008007905568498947138345, −8.599961322386130422384120738688, −8.202636679085140171560854269599, −7.00501187032476253943831094800, −6.61806322439827363777532289945, −6.14235947630945136377658789120, −4.85814194440365690532585923859, −4.16939498538761217646770198890, −2.13491463761540809783931727611, −1.26579480955164131611463823727, 1.16538185645867427812578671919, 2.72194411191581178828124957229, 3.02601349234330564814297131788, 4.23196276466849189618381349801, 5.13307210976132576910305419366, 6.17922811174685905224342143691, 7.04869897493790197191366770275, 8.423524835308421305457421874134, 9.311020851196393154328194622074, 9.983956715181401987883678359626

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