Properties

Label 2-55e2-11.2-c0-0-0
Degree $2$
Conductor $3025$
Sign $0.550 - 0.835i$
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  + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)16-s + (−0.618 + 1.90i)31-s + (0.809 − 0.587i)36-s + (0.309 + 0.951i)49-s + (1.61 + 1.17i)59-s + (0.309 − 0.951i)64-s + (0.618 + 1.90i)71-s + (−0.809 − 0.587i)81-s + 2·89-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)4-s + (−0.309 + 0.951i)9-s + (0.309 + 0.951i)16-s + (−0.618 + 1.90i)31-s + (0.809 − 0.587i)36-s + (0.309 + 0.951i)49-s + (1.61 + 1.17i)59-s + (0.309 − 0.951i)64-s + (0.618 + 1.90i)71-s + (−0.809 − 0.587i)81-s + 2·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3025\)    =    \(5^{2} \cdot 11^{2}\)
Sign: $0.550 - 0.835i$
Analytic conductor: \(1.50967\)
Root analytic conductor: \(1.22868\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3025} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3025,\ (\ :0),\ 0.550 - 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8181253189\)
\(L(\frac12)\) \(\approx\) \(0.8181253189\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.809 + 0.587i)T^{2} \)
3 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.618 - 1.90i)T + (-0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-1.61 - 1.17i)T + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 - 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 - 2T + T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T^{2} \)
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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

−8.885146444164991032101336988537, −8.481499153051907630155629885729, −7.59763942345975854902479078798, −6.75897838605572915177269212883, −5.75363140822042313008061112701, −5.20667232263036746911658833724, −4.50389218712909490844417983709, −3.55680358935366935384333542412, −2.40046962897026905977320897310, −1.25715786843295001244008191271, 0.57309192233742260054144007607, 2.25816433759928509232252633338, 3.42015113556293929284348241979, 3.90714982114245832496713604655, 4.86516480231054871234342287693, 5.70452094168977333896231709206, 6.51733483061489571459384956486, 7.41945413004145304668837427548, 8.110453514753492470841427995430, 8.822774118502955587786298482137

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