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Properties

Label 2-2205-5.4-c1-0-18
Degree $2$
Conductor $2205$
Sign $0.894 + 0.447i$
Analytic cond. $17.6070$
Root an. cond. $4.19607$
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  − 2i·2-s − 2·4-s + (−2 − i)5-s + (−2 + 4i)10-s + 3·11-s + i·13-s − 4·16-s + 7i·17-s + (4 + 2i)20-s − 6i·22-s + 6i·23-s + (3 + 4i)25-s + 2·26-s − 5·29-s − 2·31-s + 8i·32-s + ⋯
L(s)  = 1  − 1.41i·2-s − 4-s + (−0.894 − 0.447i)5-s + (−0.632 + 1.26i)10-s + 0.904·11-s + 0.277i·13-s − 16-s + 1.69i·17-s + (0.894 + 0.447i)20-s − 1.27i·22-s + 1.25i·23-s + (0.600 + 0.800i)25-s + 0.392·26-s − 0.928·29-s − 0.359·31-s + 1.41i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(17.6070\)
Root analytic conductor: \(4.19607\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2205,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.055076893\)
\(L(\frac12)\) \(\approx\) \(1.055076893\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2 + i)T \)
7 \( 1 \)
good2 \( 1 + 2iT - 2T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 - 7iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 5T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 10T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 2iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 5T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−9.185610466549716491373020878880, −8.477273678864461927943954032713, −7.60266642484173485727207332162, −6.71667020116760375332818104541, −5.70423934370409127827477889760, −4.50225760153827742349036908171, −3.84792394606019716450782592531, −3.35402035934725302253072320882, −1.91245704080113068690707135803, −1.15704039761806755584204998752, 0.41304973719446752448514052961, 2.41005433314244717753056632276, 3.55407865366965380939059433745, 4.50249991809603940604306611287, 5.22465016424296750935251716651, 6.25198238806270656338770498791, 6.86755711341955704127462141106, 7.43150220908455889732865608073, 8.068685431207513119463679726820, 8.914934077267715540340110292735

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