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Properties

Label 2-2600-104.3-c0-0-1
Degree $2$
Conductor $2600$
Sign $0.711 + 0.702i$
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  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (−0.5 + 0.866i)19-s + (0.866 + 0.499i)22-s + (−1.73 + i)23-s + 0.999·26-s + (0.866 − 0.499i)28-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (0.5 − 0.866i)9-s + (0.5 + 0.866i)11-s + (0.866 + 0.5i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s − 0.999i·18-s + (−0.5 + 0.866i)19-s + (0.866 + 0.499i)22-s + (−1.73 + i)23-s + 0.999·26-s + (0.866 − 0.499i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2600\)    =    \(2^{3} \cdot 5^{2} \cdot 13\)
Sign: $0.711 + 0.702i$
Analytic conductor: \(1.29756\)
Root analytic conductor: \(1.13910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2600} (1251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2600,\ (\ :0),\ 0.711 + 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.331297676\)
\(L(\frac12)\) \(\approx\) \(2.331297676\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-0.866 - 0.5i)T \)
good3 \( 1 + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + iT - T^{2} \)
53 \( 1 - iT - T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−9.076642343936522377521252169828, −8.278268736623450508859377964752, −7.21425173183678276256614158767, −6.50124884871405497195740788601, −5.78855115778608661062616825562, −4.96196569180102712580734685827, −3.91627357165780003260548606954, −3.70145068503795363484559927788, −1.97187022198770110473106143615, −1.61018348605316155779429868423, 1.54290568411418966785320005331, 2.66582262675062126342870991697, 3.82692314081633048224677467433, 4.41293593820350935875989440071, 5.19048447538248347690322585920, 6.07930160130108524365927013176, 6.72119592626390885985633431552, 7.66141646773502031640132174288, 8.263060163704091803249581421363, 8.683523341662305151145158447384

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