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Properties

Label 2-51e2-3.2-c0-0-2
Degree $2$
Conductor $2601$
Sign $0.577 - 0.816i$
Analytic cond. $1.29806$
Root an. cond. $1.13932$
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  + 1.41i·2-s − 1.00·4-s − 1.41i·5-s + 7-s + 2.00·10-s + 13-s + 1.41i·14-s − 0.999·16-s − 19-s + 1.41i·20-s − 1.41i·23-s − 1.00·25-s + 1.41i·26-s − 1.00·28-s + 31-s − 1.41i·32-s + ⋯
L(s)  = 1  + 1.41i·2-s − 1.00·4-s − 1.41i·5-s + 7-s + 2.00·10-s + 13-s + 1.41i·14-s − 0.999·16-s − 19-s + 1.41i·20-s − 1.41i·23-s − 1.00·25-s + 1.41i·26-s − 1.00·28-s + 31-s − 1.41i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(1.29806\)
Root analytic conductor: \(1.13932\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (2024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2601,\ (\ :0),\ 0.577 - 0.816i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.385932529\)
\(L(\frac12)\) \(\approx\) \(1.385932529\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
17 \( 1 \)
good2 \( 1 - 1.41iT - T^{2} \)
5 \( 1 + 1.41iT - T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−8.712502136809551322482288698504, −8.292449558643552495869693056336, −7.947116007536442213699005582084, −6.79603435381507656929637024601, −6.09987593422197040195441876733, −5.39370643743594992798754820461, −4.55197176692512336774066255493, −4.25150576193916439430171373150, −2.35162409662817927720561272408, −1.08862604239403277836145302538, 1.36634196148069773260660042738, 2.25768207863032055466577602759, 3.08607128049824109054955052032, 3.85152813712949199070838417329, 4.63496849065484323620311657964, 5.91583259817789952282606135466, 6.61175659698328670981637055092, 7.52170400555902544978772114909, 8.271064049761973304945799187619, 9.203325448370294046301506908560

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