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Properties

Label 2-55e2-275.244-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.994 - 0.108i$
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  + (1.80 + 0.587i)3-s − 4-s + (−0.309 − 0.951i)5-s + (2.11 + 1.53i)9-s + (−1.80 − 0.587i)12-s − 1.90i·15-s + 16-s + (0.309 + 0.951i)20-s + (1.11 − 0.363i)23-s + (−0.809 + 0.587i)25-s + (1.80 + 2.48i)27-s + (0.5 − 0.363i)31-s + (−2.11 − 1.53i)36-s + (0.809 − 2.48i)45-s + (1.80 + 0.587i)48-s + (−0.309 − 0.951i)49-s + ⋯
L(s)  = 1  + (1.80 + 0.587i)3-s − 4-s + (−0.309 − 0.951i)5-s + (2.11 + 1.53i)9-s + (−1.80 − 0.587i)12-s − 1.90i·15-s + 16-s + (0.309 + 0.951i)20-s + (1.11 − 0.363i)23-s + (−0.809 + 0.587i)25-s + (1.80 + 2.48i)27-s + (0.5 − 0.363i)31-s + (−2.11 − 1.53i)36-s + (0.809 − 2.48i)45-s + (1.80 + 0.587i)48-s + (−0.309 − 0.951i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.994 - 0.108i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (2169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.994 - 0.108i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.860244573\)
\(L(\frac12)\) \(\approx\) \(1.860244573\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.848044160967584289902791210578, −8.411498685592653022898129174308, −7.85301219042637106220899162798, −7.01689239074869558027492356230, −5.46388326470021114059101952832, −4.74898172720034316525662028302, −4.14746889958982633306099076627, −3.51103794335013787336404984706, −2.54057448576792529848031185961, −1.25044238058381565260316904405, 1.30213041208664085037404284294, 2.57265874989212559434482038634, 3.21895161699901361142566146592, 3.88803039534605373207239709529, 4.71655780342937719564987135772, 6.04667921894702722610388729400, 7.01125754401960217919196517707, 7.51264690100970916841131558573, 8.191467142663656144633980935419, 8.817149798874645284783379103081

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