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Properties

Label 2-55e2-275.19-c0-0-0
Degree $2$
Conductor $3025$
Sign $-0.555 + 0.831i$
Analytic cond. $1.50967$
Root an. cond. $1.22868$
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  + (1.11 − 1.53i)3-s + (−0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.809 − 2.48i)9-s + (−1.80 − 0.587i)12-s + (1.80 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)20-s + (1.11 − 0.363i)23-s + (0.309 + 0.951i)25-s + (−2.92 − 0.951i)27-s − 0.618·31-s + (−2.11 + 1.53i)36-s + (0.809 − 2.48i)45-s + 1.90i·48-s + (0.809 + 0.587i)49-s + ⋯
L(s)  = 1  + (1.11 − 1.53i)3-s + (−0.309 − 0.951i)4-s + (0.809 + 0.587i)5-s + (−0.809 − 2.48i)9-s + (−1.80 − 0.587i)12-s + (1.80 − 0.587i)15-s + (−0.809 + 0.587i)16-s + (0.309 − 0.951i)20-s + (1.11 − 0.363i)23-s + (0.309 + 0.951i)25-s + (−2.92 − 0.951i)27-s − 0.618·31-s + (−2.11 + 1.53i)36-s + (0.809 − 2.48i)45-s + 1.90i·48-s + (0.809 + 0.587i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.555 + 0.831i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (844, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ -0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.879420896\)
\(L(\frac12)\) \(\approx\) \(1.879420896\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T^{2} \)
3 \( 1 + (-1.11 + 1.53i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + 0.618T + T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
71 \( 1 - 0.618T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.809 + 0.587i)T^{2} \)
89 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
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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

−8.803739292219749802272622503135, −7.80287188415243124434650196702, −7.08265469238046100388997356720, −6.46698436659500119949808481687, −5.93460358966884477847137781428, −4.95425455993076305145506669245, −3.55312775187995996928537841664, −2.65750012192807084311431678891, −1.91677573095711325017573612345, −1.07947891404498147854391941835, 1.99613757702564574431658044491, 2.96928706032434438211020494019, 3.54441982917760729358255743777, 4.54872945087304162798084278841, 4.89507942836134424866654732151, 5.85532714073780142040365416942, 7.21275814867362728539143272998, 7.991857224622954921944018226087, 8.658745130289740809879553162973, 9.203630636155202949155829471048

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