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Properties

Label 2-2600-520.259-c0-0-3
Degree $2$
Conductor $2600$
Sign $0.447 + 0.894i$
Analytic cond. $1.29756$
Root an. cond. $1.13910$
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  + i·2-s − i·3-s − 4-s + 6-s − i·7-s − i·8-s + i·12-s − i·13-s + 14-s + 16-s + i·17-s − 21-s − 24-s + 26-s − i·27-s + i·28-s + ⋯
L(s)  = 1  + i·2-s − i·3-s − 4-s + 6-s − i·7-s − i·8-s + i·12-s − i·13-s + 14-s + 16-s + i·17-s − 21-s − 24-s + 26-s − i·27-s + i·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(1.29756\)
Root analytic conductor: \(1.13910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (1299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2600,\ (\ :0),\ 0.447 + 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9686447704\)
\(L(\frac12)\) \(\approx\) \(0.9686447704\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 \)
13 \( 1 + iT \)
good3 \( 1 + iT - T^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 - T^{2} \)
17 \( 1 - iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 2T + T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.575401925430477705499808454035, −7.929221259602073935583366387689, −7.28887371156706214148821550113, −6.91302117050544894852730875095, −5.95533086980217124120116354545, −5.32910483653826253440168043965, −4.14127237860610942799375375590, −3.53339647873019745774335370303, −1.87403535638343678969713244945, −0.66400420368426980752917054715, 1.62874756083702419636601617607, 2.64874601056027250164842693353, 3.52232375631695779717175110278, 4.38710127369997047439325810473, 5.01437435126731726376976657464, 5.73485165838452192604891428599, 6.90802042933717720255091797914, 7.962780757441523473655793668422, 8.965143348731152654901595631742, 9.309858786027913465307337564615

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