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Properties

Label 2-1925-385.237-c0-0-2
Degree $2$
Conductor $1925$
Sign $0.967 - 0.251i$
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  + (0.183 + 1.16i)3-s + (−0.587 − 0.809i)4-s + (−0.987 − 0.156i)7-s + (−0.363 + 0.118i)9-s + (−0.309 − 0.951i)11-s + (0.831 − 0.831i)12-s + (1.44 + 0.734i)13-s + (−0.309 + 0.951i)16-s + (0.550 − 0.280i)17-s − 1.17i·21-s + (0.329 + 0.647i)27-s + (0.453 + 0.891i)28-s + (1.53 − 1.11i)29-s + (1.04 − 0.533i)33-s + (0.309 + 0.224i)36-s + ⋯
L(s)  = 1  + (0.183 + 1.16i)3-s + (−0.587 − 0.809i)4-s + (−0.987 − 0.156i)7-s + (−0.363 + 0.118i)9-s + (−0.309 − 0.951i)11-s + (0.831 − 0.831i)12-s + (1.44 + 0.734i)13-s + (−0.309 + 0.951i)16-s + (0.550 − 0.280i)17-s − 1.17i·21-s + (0.329 + 0.647i)27-s + (0.453 + 0.891i)28-s + (1.53 − 1.11i)29-s + (1.04 − 0.533i)33-s + (0.309 + 0.224i)36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.967 - 0.251i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.967 - 0.251i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.031548533\)
\(L(\frac12)\) \(\approx\) \(1.031548533\)
\(L(1)\) 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 \( 1 + (0.987 + 0.156i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (0.587 + 0.809i)T^{2} \)
3 \( 1 + (-0.183 - 1.16i)T + (-0.951 + 0.309i)T^{2} \)
13 \( 1 + (-1.44 - 0.734i)T + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (-0.550 + 0.280i)T + (0.587 - 0.809i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.951 + 0.309i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-1.87 + 0.297i)T + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.587 + 0.809i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.0966 + 0.610i)T + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.734 - 1.44i)T + (-0.587 + 0.809i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (1.69 + 0.863i)T + (0.587 + 0.809i)T^{2} \)
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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

−9.443159334508494799518591255652, −8.897666624925229082499867629170, −8.250613687903157801972089521622, −6.81318054550359325330323782692, −6.05334407543261826142869956583, −5.43920995621225402588402188897, −4.29525389613520706607504403389, −3.84176914366833370001505620702, −2.86423610079251586881969230934, −1.01104296986437434039257663303, 1.11808705740409454760591044329, 2.57264907200381822954586261456, 3.33882046505694187898958188188, 4.28718325583763428863437718804, 5.47088078741403568386630302409, 6.41365973947871591453077751922, 7.08845433469599609082387155595, 7.80862358698006850634395678406, 8.439505584197656595428839877618, 9.122561426151113875469745106913

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