Properties

Label 2-1260-105.104-c1-0-15
Degree $2$
Conductor $1260$
Sign $-0.975 - 0.220i$
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  + (−1.08 − 1.95i)5-s + (−0.595 − 2.57i)7-s − 3.74i·11-s − 3.36·13-s − 0.841i·17-s + 5.59i·19-s − 2.35·23-s + (−2.64 + 4.24i)25-s + 1.41i·29-s + 8.66i·31-s + (−4.39 + 3.96i)35-s − 5.15i·37-s − 5.74·41-s + 3.32i·43-s − 6.43i·47-s + ⋯
L(s)  = 1  + (−0.485 − 0.874i)5-s + (−0.224 − 0.974i)7-s − 1.12i·11-s − 0.931·13-s − 0.204i·17-s + 1.28i·19-s − 0.490·23-s + (−0.529 + 0.848i)25-s + 0.262i·29-s + 1.55i·31-s + (−0.742 + 0.669i)35-s − 0.847i·37-s − 0.896·41-s + 0.507i·43-s − 0.938i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1260\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.975 - 0.220i$
Analytic conductor: \(10.0611\)
Root analytic conductor: \(3.17193\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1260} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1260,\ (\ :1/2),\ -0.975 - 0.220i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4531763094\)
\(L(\frac12)\) \(\approx\) \(0.4531763094\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 + 1.95i)T \)
7 \( 1 + (0.595 + 2.57i)T \)
good11 \( 1 + 3.74iT - 11T^{2} \)
13 \( 1 + 3.36T + 13T^{2} \)
17 \( 1 + 0.841iT - 17T^{2} \)
19 \( 1 - 5.59iT - 19T^{2} \)
23 \( 1 + 2.35T + 23T^{2} \)
29 \( 1 - 1.41iT - 29T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 + 5.15iT - 37T^{2} \)
41 \( 1 + 5.74T + 41T^{2} \)
43 \( 1 - 3.32iT - 43T^{2} \)
47 \( 1 + 6.43iT - 47T^{2} \)
53 \( 1 - 9.64T + 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 1.82iT - 67T^{2} \)
71 \( 1 + 3.74iT - 71T^{2} \)
73 \( 1 - 0.979T + 73T^{2} \)
79 \( 1 + 6.58T + 79T^{2} \)
83 \( 1 + 12.5iT - 83T^{2} \)
89 \( 1 - 2.16T + 89T^{2} \)
97 \( 1 + 12.0T + 97T^{2} \)
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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

−9.190277602105522539575686844630, −8.383194234956239049180586649649, −7.71176134666997357183995698528, −6.90752980057719474535939015867, −5.79726760331851301349088728985, −4.94609195144853364852055029607, −3.98388970645222604395035231386, −3.22015868453474389991826070062, −1.48476288833890131988092536077, −0.18718898160629468315075831989, 2.19669516295470184074007328987, 2.81775087891245844873887001937, 4.12317583147173153151675061131, 4.99220532521110805403779477205, 6.08371605212911623346550381215, 6.91587629007814984103490357269, 7.55987111078539525625763747785, 8.451898575006817097183183291723, 9.525770876160708164487890245347, 9.932741516476678044285302359383

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