Properties

Label 2-975-5.4-c1-0-22
Degree $2$
Conductor $975$
Sign $0.894 + 0.447i$
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  − 0.571i·2-s + i·3-s + 1.67·4-s + 0.571·6-s − 1.42i·7-s − 2.10i·8-s − 9-s + 3.24·11-s + 1.67i·12-s + i·13-s − 0.816·14-s + 2.14·16-s − 1.85i·17-s + 0.571i·18-s + 1.81·19-s + ⋯
L(s)  = 1  − 0.404i·2-s + 0.577i·3-s + 0.836·4-s + 0.233·6-s − 0.539i·7-s − 0.742i·8-s − 0.333·9-s + 0.978·11-s + 0.482i·12-s + 0.277i·13-s − 0.218·14-s + 0.535·16-s − 0.450i·17-s + 0.134i·18-s + 0.416·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(7.78541\)
Root analytic conductor: \(2.79023\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{975} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03911 - 0.481369i\)
\(L(\frac12)\) \(\approx\) \(2.03911 - 0.481369i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good2 \( 1 + 0.571iT - 2T^{2} \)
7 \( 1 + 1.42iT - 7T^{2} \)
11 \( 1 - 3.24T + 11T^{2} \)
17 \( 1 + 1.85iT - 17T^{2} \)
19 \( 1 - 1.81T + 19T^{2} \)
23 \( 1 + 1.52iT - 23T^{2} \)
29 \( 1 + 2.34T + 29T^{2} \)
31 \( 1 - 6.38T + 31T^{2} \)
37 \( 1 + 3.52iT - 37T^{2} \)
41 \( 1 + 3.81T + 41T^{2} \)
43 \( 1 - 10.0iT - 43T^{2} \)
47 \( 1 + 11.2iT - 47T^{2} \)
53 \( 1 - 6.81iT - 53T^{2} \)
59 \( 1 - 5.91T + 59T^{2} \)
61 \( 1 - 5.48T + 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + 9.81T + 71T^{2} \)
73 \( 1 - 5.32iT - 73T^{2} \)
79 \( 1 - 2.96T + 79T^{2} \)
83 \( 1 - 3.14iT - 83T^{2} \)
89 \( 1 - 4.85T + 89T^{2} \)
97 \( 1 + 1.51iT - 97T^{2} \)
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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

−10.03810576247942220534989730191, −9.382086198047835116542690149953, −8.370573363103408041824950722523, −7.25068968151920138871124285160, −6.64943086646675074833544723145, −5.64905931520331832370922769751, −4.38554711460991127668667991895, −3.61277829827314489192312517885, −2.54028078774760268499247371892, −1.14156657728844258813537225824, 1.40161473094106842375430269217, 2.49551731674226799527677491131, 3.61214173654488836065981682241, 5.13194639966016353169924075034, 6.03230847919906206532415135147, 6.61865104457482719747928520138, 7.46717136874755700162035282537, 8.250817509673524691321594265795, 9.033352703147176805330207870242, 10.08932111293048838132317010449

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