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Properties

Label 2-55e2-275.134-c0-0-0
Degree $2$
Conductor $3025$
Sign $-0.650 + 0.759i$
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  + (−0.690 + 0.951i)3-s + (−0.309 + 0.951i)4-s + (−0.309 − 0.951i)5-s + (−0.118 − 0.363i)9-s + (−0.690 − 0.951i)12-s + (1.11 + 0.363i)15-s + (−0.809 − 0.587i)16-s + 0.999·20-s + (−1.11 + 1.53i)23-s + (−0.809 + 0.587i)25-s + (−0.690 − 0.224i)27-s + (−1.30 − 0.951i)31-s + 0.381·36-s + (−0.309 + 0.224i)45-s + (1.11 − 0.363i)48-s + (−0.309 − 0.951i)49-s + ⋯
L(s)  = 1  + (−0.690 + 0.951i)3-s + (−0.309 + 0.951i)4-s + (−0.309 − 0.951i)5-s + (−0.118 − 0.363i)9-s + (−0.690 − 0.951i)12-s + (1.11 + 0.363i)15-s + (−0.809 − 0.587i)16-s + 0.999·20-s + (−1.11 + 1.53i)23-s + (−0.809 + 0.587i)25-s + (−0.690 − 0.224i)27-s + (−1.30 − 0.951i)31-s + 0.381·36-s + (−0.309 + 0.224i)45-s + (1.11 − 0.363i)48-s + (−0.309 − 0.951i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.650 + 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $-0.650 + 0.759i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ -0.650 + 0.759i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1083632109\)
\(L(\frac12)\) \(\approx\) \(0.1083632109\)
\(L(1)\) 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 \( 1 + (0.309 - 0.951i)T^{2} \)
3 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 - 1.17iT - T^{2} \)
59 \( 1 + 1.61T + T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + 1.17iT - T^{2} \)
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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

−9.385140111708582683122500301583, −8.764363049546262257624259151678, −7.74764855120628535142020107541, −7.54499759448878853062910119630, −6.08179068618926777076000322369, −5.37946466892696799977535997463, −4.65807008354912184241652705504, −4.01394552617897226295014044890, −3.40923656612802608303364718945, −1.86584868902776272978642789766, 0.07313577850109593514189779005, 1.47143035330809889023616140535, 2.41927002268645371008590353225, 3.69083744084348292100948120497, 4.64483841889728989211317797551, 5.64022136723813100716823646983, 6.26745767464469866956821490113, 6.73367068092158064656986294309, 7.45923456809587524803348411761, 8.305979936491520022711841492029

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