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Properties

Label 2-2325-5.4-c1-0-44
Degree $2$
Conductor $2325$
Sign $-0.894 - 0.447i$
Analytic cond. $18.5652$
Root an. cond. $4.30873$
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  + 2.51i·2-s + i·3-s − 4.32·4-s − 2.51·6-s + 0.485i·7-s − 5.83i·8-s − 9-s + 5.02·11-s − 4.32i·12-s − 3.51i·13-s − 1.22·14-s + 6.02·16-s − 1.32i·17-s − 2.51i·18-s + 6.64·19-s + ⋯
L(s)  = 1  + 1.77i·2-s + 0.577i·3-s − 2.16·4-s − 1.02·6-s + 0.183i·7-s − 2.06i·8-s − 0.333·9-s + 1.51·11-s − 1.24i·12-s − 0.974i·13-s − 0.326·14-s + 1.50·16-s − 0.320i·17-s − 0.592i·18-s + 1.52·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2325\)    =    \(3 \cdot 5^{2} \cdot 31\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(18.5652\)
Root analytic conductor: \(4.30873\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2325} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2325,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.762409053\)
\(L(\frac12)\) \(\approx\) \(1.762409053\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
31 \( 1 - T \)
good2 \( 1 - 2.51iT - 2T^{2} \)
7 \( 1 - 0.485iT - 7T^{2} \)
11 \( 1 - 5.02T + 11T^{2} \)
13 \( 1 + 3.51iT - 13T^{2} \)
17 \( 1 + 1.32iT - 17T^{2} \)
19 \( 1 - 6.64T + 19T^{2} \)
23 \( 1 + 0.292iT - 23T^{2} \)
29 \( 1 - 9.86T + 29T^{2} \)
37 \( 1 - 5.51iT - 37T^{2} \)
41 \( 1 + 7.02T + 41T^{2} \)
43 \( 1 - 1.02iT - 43T^{2} \)
47 \( 1 + 6.93iT - 47T^{2} \)
53 \( 1 - 1.70iT - 53T^{2} \)
59 \( 1 - 2.19T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 9.12iT - 67T^{2} \)
71 \( 1 - 13.4T + 71T^{2} \)
73 \( 1 + 12.5iT - 73T^{2} \)
79 \( 1 - 0.349T + 79T^{2} \)
83 \( 1 + 10.9iT - 83T^{2} \)
89 \( 1 + 5.03T + 89T^{2} \)
97 \( 1 - 10.4iT - 97T^{2} \)
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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

−9.059184677312886187266293385027, −8.470888760186292909042615746343, −7.76202070956464440800460188600, −6.86170095935949362604575473609, −6.33760363833248444027085729729, −5.41862144675286811380922963762, −4.90547222480100975117330826243, −3.96321592576214188544464741894, −3.05245410288999508938190012725, −0.921492907264923545059491545631, 0.940233173126443279942010072190, 1.58191383953626723770369112874, 2.64339026572851748275400421342, 3.60514296981998117455827246718, 4.24386445488835475647504558918, 5.20112615498392525652251350388, 6.40929081763163814939051928925, 7.08318969208310867726577281031, 8.243365398964764626504683484659, 8.974056902381879693702778799479

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