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Properties

Label 2-2275-91.55-c0-0-1
Degree $2$
Conductor $2275$
Sign $-0.252 + 0.967i$
Analytic cond. $1.13537$
Root an. cond. $1.06553$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)7-s + (−1 − 1.73i)11-s − 0.999i·12-s + (0.866 + 0.5i)13-s + (−0.499 − 0.866i)16-s + (0.866 + 0.5i)17-s − 0.999·21-s + i·27-s + (−0.866 + 0.499i)28-s + (−0.5 − 0.866i)29-s + (−1.73 − 0.999i)33-s + 0.999·39-s − 1.99·44-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)7-s + (−1 − 1.73i)11-s − 0.999i·12-s + (0.866 + 0.5i)13-s + (−0.499 − 0.866i)16-s + (0.866 + 0.5i)17-s − 0.999·21-s + i·27-s + (−0.866 + 0.499i)28-s + (−0.5 − 0.866i)29-s + (−1.73 − 0.999i)33-s + 0.999·39-s − 1.99·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2275\)    =    \(5^{2} \cdot 7 \cdot 13\)
Sign: $-0.252 + 0.967i$
Analytic conductor: \(1.13537\)
Root analytic conductor: \(1.06553\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2275} (601, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2275,\ (\ :0),\ -0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.507494665\)
\(L(\frac12)\) \(\approx\) \(1.507494665\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
good2 \( 1 + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.896778262540711678336004266261, −8.171934237046287380646346909351, −7.54913786389520965735872968421, −6.58445223628978571532450528773, −5.95281107030051150163709685796, −5.30483658264066019790464244060, −3.74254715573253793787104950830, −3.09447290938680458123310773301, −2.16020458520764146925778402145, −0.917210383352429555871925644492, 2.07074358118097646865450519510, 2.96597660109454638267225872350, 3.40184242105267454221486313404, 4.40356331725902045997403359452, 5.48033521056950563723318170102, 6.44887722338106571734020941540, 7.33928654371640213226856507252, 7.87893133388850650490544853602, 8.690610003056477988203212775909, 9.376584121251505327317965238352

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