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Properties

Label 2-55e2-275.194-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.920 - 0.390i$
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 + (0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (2.11 − 1.53i)9-s + (−1.80 − 0.587i)12-s + (−1.11 + 1.53i)15-s + (0.309 + 0.951i)16-s + 20-s + (1.11 − 0.363i)23-s + (0.309 − 0.951i)25-s + (−1.80 + 2.48i)27-s + (−0.190 − 0.587i)31-s + 2.61·36-s + (0.809 − 2.48i)45-s + (−1.11 − 1.53i)48-s + (0.809 − 0.587i)49-s + ⋯
L(s)  = 1  + (−1.80 + 0.587i)3-s + (0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (2.11 − 1.53i)9-s + (−1.80 − 0.587i)12-s + (−1.11 + 1.53i)15-s + (0.309 + 0.951i)16-s + 20-s + (1.11 − 0.363i)23-s + (0.309 − 0.951i)25-s + (−1.80 + 2.48i)27-s + (−0.190 − 0.587i)31-s + 2.61·36-s + (0.809 − 2.48i)45-s + (−1.11 − 1.53i)48-s + (0.809 − 0.587i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.920 - 0.390i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (2944, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.021470931\)
\(L(\frac12)\) \(\approx\) \(1.021470931\)
\(L(1)\) 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 \( 1 + (-0.809 - 0.587i)T^{2} \)
3 \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 - 0.951i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + 1.90iT - T^{2} \)
59 \( 1 - 0.618T + T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 - 1.90iT - T^{2} \)
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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

−9.155143928416215208313467209183, −8.194518014435923511013033596747, −7.03358260413250359424756240371, −6.62624240007741119522825578146, −5.82601724116125751635753671949, −5.26923726817951853709165031747, −4.49133141724817901330404197009, −3.58455354127411128248095001667, −2.21053399474756188033869874953, −1.00725800935142357667572242877, 1.10262673677885935778487746689, 1.88177800519145280098475955828, 2.96570325133244635484215960685, 4.58741173107712093157818629065, 5.42032442802852276060673007593, 5.85505446850247514680282188981, 6.48662930086542364719028759300, 7.11957897918104013827063648448, 7.52865078552238114127518197687, 9.066056858300024746669864373389

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