Properties

Label 2-1260-5.4-c1-0-9
Degree $2$
Conductor $1260$
Sign $0.447 + 0.894i$
Analytic cond. $10.0611$
Root an. cond. $3.17193$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2 + i)5-s + i·7-s − 4·11-s − 2i·13-s − 2i·17-s + 2·19-s − 6i·23-s + (3 − 4i)25-s + 6·29-s + 6·31-s + (−1 − 2i)35-s − 4i·37-s + 4i·43-s − 4i·47-s − 49-s + ⋯
L(s)  = 1  + (−0.894 + 0.447i)5-s + 0.377i·7-s − 1.20·11-s − 0.554i·13-s − 0.485i·17-s + 0.458·19-s − 1.25i·23-s + (0.600 − 0.800i)25-s + 1.11·29-s + 1.07·31-s + (−0.169 − 0.338i)35-s − 0.657i·37-s + 0.609i·43-s − 0.583i·47-s − 0.142·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9671445661\)
\(L(\frac12)\) \(\approx\) \(0.9671445661\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (2 - i)T \)
7 \( 1 - iT \)
good11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 4iT - 47T^{2} \)
53 \( 1 + 2iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 14iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 16iT - 83T^{2} \)
89 \( 1 - 16T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−9.648307011387804568427389851268, −8.424041671504605213513899162462, −8.054098069079193760926880822262, −7.16116912953966265724109352289, −6.30159369237976561777077681965, −5.20569091422803011917605265448, −4.45778000480376715795849015311, −3.15298669682829436029692126461, −2.53401078227379528118069123077, −0.47043561184252440048275647383, 1.14909337484002361877807502805, 2.75155835323988551871707842161, 3.80552369448707893279649198732, 4.68324751356373303469515082095, 5.47333615347711159847509444142, 6.66379707164328041190353857707, 7.55514447012059832831887646127, 8.106167573825536348188629952611, 8.891353944728336320666834064006, 9.924142714344460158808893383846

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