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Properties

Label 2-1925-385.118-c0-0-5
Degree $2$
Conductor $1925$
Sign $0.748 - 0.662i$
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  + (1.93 + 0.306i)2-s + (2.68 + 0.873i)4-s + (−0.891 + 0.453i)7-s + (3.18 + 1.62i)8-s + (−0.587 − 0.809i)9-s + (−0.104 + 0.994i)11-s + (−1.86 + 0.604i)14-s + (3.36 + 2.44i)16-s + (−0.888 − 1.74i)18-s + (−0.506 + 1.88i)22-s + (−1.05 − 1.05i)23-s + (−2.79 + 0.442i)28-s + (0.614 − 1.89i)29-s + (3.23 + 3.23i)32-s + (−0.873 − 2.68i)36-s + (−0.188 − 0.370i)37-s + ⋯
L(s)  = 1  + (1.93 + 0.306i)2-s + (2.68 + 0.873i)4-s + (−0.891 + 0.453i)7-s + (3.18 + 1.62i)8-s + (−0.587 − 0.809i)9-s + (−0.104 + 0.994i)11-s + (−1.86 + 0.604i)14-s + (3.36 + 2.44i)16-s + (−0.888 − 1.74i)18-s + (−0.506 + 1.88i)22-s + (−1.05 − 1.05i)23-s + (−2.79 + 0.442i)28-s + (0.614 − 1.89i)29-s + (3.23 + 3.23i)32-s + (−0.873 − 2.68i)36-s + (−0.188 − 0.370i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.748 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.748 - 0.662i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.748 - 0.662i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.331841129\)
\(L(\frac12)\) \(\approx\) \(3.331841129\)
\(L(1)\) 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 \( 1 + (0.891 - 0.453i)T \)
11 \( 1 + (0.104 - 0.994i)T \)
good2 \( 1 + (-1.93 - 0.306i)T + (0.951 + 0.309i)T^{2} \)
3 \( 1 + (0.587 + 0.809i)T^{2} \)
13 \( 1 + (0.951 + 0.309i)T^{2} \)
17 \( 1 + (0.951 - 0.309i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (1.05 + 1.05i)T + iT^{2} \)
29 \( 1 + (-0.614 + 1.89i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.188 + 0.370i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (0.147 - 0.147i)T - iT^{2} \)
47 \( 1 + (-0.587 - 0.809i)T^{2} \)
53 \( 1 + (1.16 + 0.183i)T + (0.951 + 0.309i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T^{2} \)
67 \( 1 + (0.575 - 0.575i)T - iT^{2} \)
71 \( 1 + (-0.169 - 0.122i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.658 - 0.478i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.951 + 0.309i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.951 + 0.309i)T^{2} \)
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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

−9.650063904473410572283725691660, −8.440176983721956206750917290235, −7.58443380607511809925174323744, −6.59179753257210675333889283591, −6.26323211091633919204298800069, −5.52258801030442495968202749726, −4.49913874707542323231780972729, −3.85021796216280967126905713236, −2.86987709642949165984141351290, −2.20191771326094293604793026826, 1.65059709486750275026527209005, 2.97227315264438187311900354340, 3.32846676733113639676177370144, 4.34056244840584158492498065998, 5.28283293751579340927980588517, 5.85172588852198234308093919967, 6.59287056238995813700093543948, 7.38303718943146621869792993536, 8.280270645243739026799309453319, 9.583311405876621947376749956686

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