Properties

Label 2-975-65.64-c1-0-39
Degree $2$
Conductor $975$
Sign $-0.983 + 0.181i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
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  + 1.73·2-s i·3-s + 0.999·4-s − 1.73i·6-s − 3.46·7-s − 1.73·8-s − 9-s − 3.46i·11-s − 0.999i·12-s + (−3.46 − i)13-s − 5.99·14-s − 5·16-s + 6i·17-s − 1.73·18-s − 3.46i·19-s + ⋯
L(s)  = 1  + 1.22·2-s − 0.577i·3-s + 0.499·4-s − 0.707i·6-s − 1.30·7-s − 0.612·8-s − 0.333·9-s − 1.04i·11-s − 0.288i·12-s + (−0.960 − 0.277i)13-s − 1.60·14-s − 1.25·16-s + 1.45i·17-s − 0.408·18-s − 0.794i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.983 + 0.181i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.983 + 0.181i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0771142 - 0.842218i\)
\(L(\frac12)\) \(\approx\) \(0.0771142 - 0.842218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 \)
13 \( 1 + (3.46 + i)T \)
good2 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 + 3.46T + 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 - 6.92T + 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 10.3T + 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 3.46T + 83T^{2} \)
89 \( 1 - 6.92iT - 89T^{2} \)
97 \( 1 + 13.8T + 97T^{2} \)
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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

−9.521699422454859379532961752886, −8.853945936814851135550603665353, −7.73768224497396035369408887935, −6.71758931145626254154663823348, −6.03327563681204834057041476941, −5.44460207615408222237527273958, −4.11200158860027584532521766391, −3.30481633812336266518561847930, −2.42403878170011824971894030886, −0.23834967469095542627178599101, 2.50983029554224408163554570570, 3.30448502977368861323606431013, 4.31351501756250987101297853367, 4.97226241334813494253518490113, 5.90588513756039764409952196647, 6.77923518745790684069354885021, 7.59608130729683496678090199270, 9.205972085610464597568657338929, 9.525107082714969397378188780982, 10.23012343234941444732000997408

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