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Properties

Label 2-1925-385.279-c0-0-5
Degree $2$
Conductor $1925$
Sign $-0.957 + 0.288i$
Analytic cond. $0.960700$
Root an. cond. $0.980153$
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  + (1.14 − 1.58i)2-s + (−0.873 − 2.68i)4-s + (0.951 − 0.309i)7-s + (−3.39 − 1.10i)8-s + (0.809 + 0.587i)9-s + (−0.104 − 0.994i)11-s + (0.604 − 1.86i)14-s + (−3.36 + 2.44i)16-s + (1.86 − 0.604i)18-s + (−1.69 − 0.978i)22-s + 1.33i·23-s + (−1.66 − 2.28i)28-s + (0.0646 + 0.198i)29-s + 4.57i·32-s + (0.873 − 2.68i)36-s + (−1.86 + 0.604i)37-s + ⋯
L(s)  = 1  + (1.14 − 1.58i)2-s + (−0.873 − 2.68i)4-s + (0.951 − 0.309i)7-s + (−3.39 − 1.10i)8-s + (0.809 + 0.587i)9-s + (−0.104 − 0.994i)11-s + (0.604 − 1.86i)14-s + (−3.36 + 2.44i)16-s + (1.86 − 0.604i)18-s + (−1.69 − 0.978i)22-s + 1.33i·23-s + (−1.66 − 2.28i)28-s + (0.0646 + 0.198i)29-s + 4.57i·32-s + (0.873 − 2.68i)36-s + (−1.86 + 0.604i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.957 + 0.288i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ -0.957 + 0.288i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.120060736\)
\(L(\frac12)\) \(\approx\) \(2.120060736\)
\(L(1)\) 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 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
good2 \( 1 + (-1.14 + 1.58i)T + (-0.309 - 0.951i)T^{2} \)
3 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.33iT - T^{2} \)
29 \( 1 + (-0.0646 - 0.198i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (1.86 - 0.604i)T + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + 0.209iT - T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + 1.82iT - T^{2} \)
71 \( 1 + (-0.169 + 0.122i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.47 - 1.07i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.309 + 0.951i)T^{2} \)
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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

−9.374639114522056340912302610833, −8.474257704917738853712390811701, −7.44170623782081425244374628055, −6.28995094566477565266326938207, −5.28881200604761983804291672028, −4.88629776426640206879854134222, −3.91788079791182720730223933379, −3.22020840021047724000568431477, −1.98848461305532804372159372989, −1.25356020293228851841569363471, 2.16987635669850160132335701699, 3.50850022085477171616402410963, 4.42943366007254075791561187572, 4.85296218549655100042368197063, 5.72334047690203782581525661309, 6.69285567007733964220204276806, 7.12045479539012419284486820304, 7.943207603108455008212841379840, 8.596691756640877274503853966851, 9.365426776933709853625306614940

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