↑

Properties

Label 2-2300-115.68-c1-0-11
Degree $2$
Conductor $2300$
Sign $0.127 - 0.991i$
Analytic cond. $18.3655$
Root an. cond. $4.28550$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.84 + 1.84i)3-s + (−2.81 − 2.81i)7-s + 3.78i·9-s + 0.669i·11-s + (1.35 + 1.35i)13-s + (2.92 + 2.92i)17-s + 5.33·19-s − 10.3i·21-s + (2.40 + 4.14i)23-s + (−1.43 + 1.43i)27-s + 7.95i·29-s − 0.930·31-s + (−1.23 + 1.23i)33-s + (−6.08 − 6.08i)37-s + 4.97i·39-s + ⋯
L(s)  = 1  + (1.06 + 1.06i)3-s + (−1.06 − 1.06i)7-s + 1.26i·9-s + 0.201i·11-s + (0.374 + 0.374i)13-s + (0.708 + 0.708i)17-s + 1.22·19-s − 2.26i·21-s + (0.502 + 0.864i)23-s + (−0.276 + 0.276i)27-s + 1.47i·29-s − 0.167·31-s + (−0.214 + 0.214i)33-s + (−1.00 − 1.00i)37-s + 0.797i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.127 - 0.991i$
Analytic conductor: \(18.3655\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (1793, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :1/2),\ 0.127 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.347761097\)
\(L(\frac12)\) \(\approx\) \(2.347761097\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 + (-2.40 - 4.14i)T \)
good3 \( 1 + (-1.84 - 1.84i)T + 3iT^{2} \)
7 \( 1 + (2.81 + 2.81i)T + 7iT^{2} \)
11 \( 1 - 0.669iT - 11T^{2} \)
13 \( 1 + (-1.35 - 1.35i)T + 13iT^{2} \)
17 \( 1 + (-2.92 - 2.92i)T + 17iT^{2} \)
19 \( 1 - 5.33T + 19T^{2} \)
29 \( 1 - 7.95iT - 29T^{2} \)
31 \( 1 + 0.930T + 31T^{2} \)
37 \( 1 + (6.08 + 6.08i)T + 37iT^{2} \)
41 \( 1 - 3.17T + 41T^{2} \)
43 \( 1 + (-6.02 + 6.02i)T - 43iT^{2} \)
47 \( 1 + (0.701 - 0.701i)T - 47iT^{2} \)
53 \( 1 + (6.13 - 6.13i)T - 53iT^{2} \)
59 \( 1 - 6.43iT - 59T^{2} \)
61 \( 1 - 14.7iT - 61T^{2} \)
67 \( 1 + (-2.49 - 2.49i)T + 67iT^{2} \)
71 \( 1 - 0.918T + 71T^{2} \)
73 \( 1 + (6.06 + 6.06i)T + 73iT^{2} \)
79 \( 1 - 8.35T + 79T^{2} \)
83 \( 1 + (8.60 - 8.60i)T - 83iT^{2} \)
89 \( 1 - 1.23T + 89T^{2} \)
97 \( 1 + (-4.61 - 4.61i)T + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.176150351822782854553627354821, −8.756213562708015750882966326121, −7.49288191963688085669678500454, −7.22363058152833804447130036556, −5.99930497617630220322918399068, −5.06040271453446875628279096390, −3.92171226280930153682529382533, −3.61420526806370818293926170236, −2.85222093232581373322953858912, −1.28109537906094329140921480363, 0.77188751781037577758878225895, 2.10640034034083557749350237405, 3.02732917435669301141418400679, 3.32909831075067967443362218644, 4.96193213592176687939138388242, 5.94490969547685620044841120916, 6.54613914974755113842123787112, 7.41244837276562027657898894434, 8.085363360889540128820178894894, 8.718648860980926707381968477625

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