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Properties

Label 2-1925-385.118-c0-0-6
Degree $2$
Conductor $1925$
Sign $0.948 + 0.317i$
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  + (1.32 + 0.209i)2-s + (0.752 + 0.244i)4-s + (0.891 − 0.453i)7-s + (−0.249 − 0.127i)8-s + (−0.587 − 0.809i)9-s + (0.913 − 0.406i)11-s + (1.27 − 0.413i)14-s + (−0.942 − 0.684i)16-s + (−0.607 − 1.19i)18-s + (1.29 − 0.346i)22-s + (0.294 + 0.294i)23-s + (0.781 − 0.123i)28-s + (−0.251 + 0.773i)29-s + (−0.904 − 0.904i)32-s + (−0.244 − 0.752i)36-s + (0.674 + 1.32i)37-s + ⋯
L(s)  = 1  + (1.32 + 0.209i)2-s + (0.752 + 0.244i)4-s + (0.891 − 0.453i)7-s + (−0.249 − 0.127i)8-s + (−0.587 − 0.809i)9-s + (0.913 − 0.406i)11-s + (1.27 − 0.413i)14-s + (−0.942 − 0.684i)16-s + (−0.607 − 1.19i)18-s + (1.29 − 0.346i)22-s + (0.294 + 0.294i)23-s + (0.781 − 0.123i)28-s + (−0.251 + 0.773i)29-s + (−0.904 − 0.904i)32-s + (−0.244 − 0.752i)36-s + (0.674 + 1.32i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.948 + 0.317i$
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: \((2,\ 1925,\ (\ :0),\ 0.948 + 0.317i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.303221501\)
\(L(\frac12)\) \(\approx\) \(2.303221501\)
\(L(1)\) 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 \( 1 + (-0.913 + 0.406i)T \)
good2 \( 1 + (-1.32 - 0.209i)T + (0.951 + 0.309i)T^{2} \)
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 \( 1 + (-0.294 - 0.294i)T + iT^{2} \)
29 \( 1 + (0.251 - 0.773i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.674 - 1.32i)T + (-0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + (1.29 - 1.29i)T - iT^{2} \)
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 \( 1 + (1.40 - 1.40i)T - iT^{2} \)
71 \( 1 + (1.47 + 1.07i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-1.60 + 1.16i)T + (0.309 - 0.951i)T^{2} \)
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

−9.172263317052385840239946581904, −8.634918276194467872628432186323, −7.57635243915129198563799976739, −6.65637399713930571704934578997, −6.10547678393937115464829896183, −5.21483692702187133104617914781, −4.46759561518035987751787027542, −3.66246494407851001405625318380, −2.93007799161321929532019701686, −1.29606814374758239503916309806, 1.88416072870360442121743020042, 2.62687064081789340760078182198, 3.82241802135729436066776292237, 4.54137732660491521668722816816, 5.30438917104278109051575960758, 5.87715777059928493535501712372, 6.86347496148800268100799390752, 7.85254941200406929769063500125, 8.665449609887634295505175043766, 9.271548906256412708496023688371

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