Properties

Label 2-1925-385.118-c0-0-0
Degree $2$
Conductor $1925$
Sign $0.161 - 0.986i$
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  + (−0.610 − 0.0966i)2-s + (−0.587 − 0.190i)4-s + (−0.891 + 0.453i)7-s + (0.891 + 0.453i)8-s + (−0.587 − 0.809i)9-s + (−0.809 − 0.587i)11-s + (0.587 − 0.190i)14-s + (0.280 + 0.550i)18-s + (0.437 + 0.437i)22-s + (1.34 + 1.34i)23-s + (0.610 − 0.0966i)28-s + (−0.363 + 1.11i)29-s + (−0.707 − 0.707i)32-s + (0.190 + 0.587i)36-s + (0.863 + 1.69i)37-s + ⋯
L(s)  = 1  + (−0.610 − 0.0966i)2-s + (−0.587 − 0.190i)4-s + (−0.891 + 0.453i)7-s + (0.891 + 0.453i)8-s + (−0.587 − 0.809i)9-s + (−0.809 − 0.587i)11-s + (0.587 − 0.190i)14-s + (0.280 + 0.550i)18-s + (0.437 + 0.437i)22-s + (1.34 + 1.34i)23-s + (0.610 − 0.0966i)28-s + (−0.363 + 1.11i)29-s + (−0.707 − 0.707i)32-s + (0.190 + 0.587i)36-s + (0.863 + 1.69i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.161 - 0.986i$
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: \((2,\ 1925,\ (\ :0),\ 0.161 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3599549390\)
\(L(\frac12)\) \(\approx\) \(0.3599549390\)
\(L(1)\) 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 \( 1 + (0.809 + 0.587i)T \)
good2 \( 1 + (0.610 + 0.0966i)T + (0.951 + 0.309i)T^{2} \)
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 \( 1 + (-1.34 - 1.34i)T + iT^{2} \)
29 \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.863 - 1.69i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (1.14 - 1.14i)T - iT^{2} \)
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 \( 1 + (0.831 - 0.831i)T - iT^{2} \)
71 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \)
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

−9.607215188501490530828669291452, −8.801748157107014375859397855190, −8.337422491097386798096923120559, −7.32847102367528182692778088599, −6.36778819487240129258938674626, −5.55855826543431378279005114195, −4.87856936717468442311399743903, −3.47372277478618093175476287492, −2.89690355618453304309813962752, −1.20732778528109601876311556512, 0.36622070047022712646765634837, 2.22776049909773345645613158366, 3.27052711034396073534136605642, 4.38929484749259942848587886952, 5.05143533214386780399732484633, 6.10718517956591315864772190750, 7.16934916111430154650508667079, 7.68133463496012051151796162099, 8.479065059966077480084221003678, 9.199616318619507195315356206136

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