Properties

Label 2-1925-385.153-c0-0-1
Degree $2$
Conductor $1925$
Sign $0.981 + 0.189i$
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  + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + i·9-s + (−0.5 + 0.866i)11-s + 1.00i·14-s + 1.00·16-s + (0.707 − 0.707i)18-s + (0.965 − 0.258i)22-s + (1.22 + 1.22i)23-s + 1.73·29-s + (1.22 − 1.22i)37-s + (−0.707 + 0.707i)43-s − 1.73i·46-s + 1.00i·49-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + (−0.707 − 0.707i)7-s + (−0.707 + 0.707i)8-s + i·9-s + (−0.5 + 0.866i)11-s + 1.00i·14-s + 1.00·16-s + (0.707 − 0.707i)18-s + (0.965 − 0.258i)22-s + (1.22 + 1.22i)23-s + 1.73·29-s + (1.22 − 1.22i)37-s + (−0.707 + 0.707i)43-s − 1.73i·46-s + 1.00i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 + 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.981 + 0.189i$
Analytic conductor: \(0.960700\)
Root analytic conductor: \(0.980153\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1693, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1925,\ (\ :0),\ 0.981 + 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6304867283\)
\(L(\frac12)\) \(\approx\) \(0.6304867283\)
\(L(1)\) 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 \( 1 + (0.707 + 0.707i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.707 + 0.707i)T + iT^{2} \)
3 \( 1 - iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
29 \( 1 - 1.73T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
71 \( 1 + T + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - 1.73T + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−9.590750636608045121463550029526, −8.807129803057370437094631628463, −7.82439911285432407767421210151, −7.25815674562521146960369252099, −6.23986686428447947806892639301, −5.24319515473300988455971933639, −4.48949623889906382995447803777, −3.14747836967154813641013206123, −2.35071810926086239450272373274, −1.16301696950919786600312996674, 0.68376718693344766621465757397, 2.86074608424259575002916647352, 3.24829722997517536025698205217, 4.59863582107155128386829971960, 5.81054437118105139804442000036, 6.45741181010360322582296269897, 6.89473788628772646164827166542, 8.091364174883814501364510159606, 8.605064687381194944070593015379, 9.164745529753125658413340534736

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