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Properties

Label 2-51e2-17.5-c0-0-1
Degree $2$
Conductor $2601$
Sign $0.855 + 0.518i$
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  + (0.707 − 0.707i)4-s + (1.41 + 1.41i)13-s − 1.00i·16-s + (−0.765 − 1.84i)19-s + (0.923 + 0.382i)25-s + (−0.765 + 1.84i)43-s + (0.923 − 0.382i)49-s + 2.00·52-s + (−0.707 − 0.707i)64-s − 2i·67-s + (−1.84 − 0.765i)76-s + (0.923 − 0.382i)100-s − 2·103-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)4-s + (1.41 + 1.41i)13-s − 1.00i·16-s + (−0.765 − 1.84i)19-s + (0.923 + 0.382i)25-s + (−0.765 + 1.84i)43-s + (0.923 − 0.382i)49-s + 2.00·52-s + (−0.707 − 0.707i)64-s − 2i·67-s + (−1.84 − 0.765i)76-s + (0.923 − 0.382i)100-s − 2·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2601\)    =    \(3^{2} \cdot 17^{2}\)
Sign: $0.855 + 0.518i$
Analytic conductor: \(1.29806\)
Root analytic conductor: \(1.13932\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2601} (1603, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2601,\ (\ :0),\ 0.855 + 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.520063880\)
\(L(\frac12)\) \(\approx\) \(1.520063880\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T^{2} \)
5 \( 1 + (-0.923 - 0.382i)T^{2} \)
7 \( 1 + (-0.923 + 0.382i)T^{2} \)
11 \( 1 + (0.382 + 0.923i)T^{2} \)
13 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
19 \( 1 + (0.765 + 1.84i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (-0.382 - 0.923i)T^{2} \)
29 \( 1 + (0.923 + 0.382i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (-0.923 + 0.382i)T^{2} \)
43 \( 1 + (0.765 - 1.84i)T + (-0.707 - 0.707i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (-0.707 + 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (0.923 - 0.382i)T^{2} \)
67 \( 1 + 2iT - T^{2} \)
71 \( 1 + (-0.382 + 0.923i)T^{2} \)
73 \( 1 + (-0.923 - 0.382i)T^{2} \)
79 \( 1 + (0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (0.923 + 0.382i)T^{2} \)
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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

−9.065279142806858595409497730641, −8.379277371170779751455966277518, −7.20332676104790212318031286470, −6.60704510487934772993974887091, −6.17578798964012096030241220577, −5.05645519148939025475927251862, −4.35688827590659049073272611394, −3.17537983926572875901218284847, −2.17041958843222043484671134541, −1.18428138342087438530047280525, 1.38791059031484716179701723393, 2.56461151376582762569598950178, 3.51138279366475648569192805146, 4.03790781102642211384386272791, 5.47729664095191410574856390056, 6.05961125710296000992590487627, 6.84135339757116892587326158792, 7.72206039990709787772000488026, 8.392882514025795792229274024718, 8.723955525775740863940634459629

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