Properties

Label 2-975-5.4-c1-0-32
Degree $2$
Conductor $975$
Sign $-0.447 + 0.894i$
Analytic cond. $7.78541$
Root an. cond. $2.79023$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s i·3-s − 2·4-s + 2·6-s − 3i·7-s − 9-s − 5·11-s + 2i·12-s i·13-s + 6·14-s − 4·16-s + 5i·17-s − 2i·18-s − 2·19-s − 3·21-s − 10i·22-s + ⋯
L(s)  = 1  + 1.41i·2-s − 0.577i·3-s − 4-s + 0.816·6-s − 1.13i·7-s − 0.333·9-s − 1.50·11-s + 0.577i·12-s − 0.277i·13-s + 1.60·14-s − 16-s + 1.21i·17-s − 0.471i·18-s − 0.458·19-s − 0.654·21-s − 2.13i·22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(975\)    =    \(3 \cdot 5^{2} \cdot 13\)
Sign: $-0.447 + 0.894i$
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: \(1\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 2T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 5T + 11T^{2} \)
17 \( 1 - 5iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 3iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 10iT - 47T^{2} \)
53 \( 1 + 9iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 11T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 8iT - 83T^{2} \)
89 \( 1 - 11T + 89T^{2} \)
97 \( 1 + 9iT - 97T^{2} \)
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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

−9.541758690289089937798999864926, −8.386119843524000244102713291511, −7.79724243594435474886350774139, −7.35312019626074465076408138506, −6.41671683791534006534281028614, −5.65727426264097005904742242830, −4.78772740075501855139012978932, −3.55738854060454395267998035213, −1.98095357619638911834050674061, 0, 2.08960137747437871260749079517, 2.74318712192696589720572087573, 3.72774110639731016572210183479, 4.95410475736013450010420585573, 5.52219927950668028705023352890, 6.91040422787831570441579040453, 8.097372236858610332846946511579, 9.070956808484618678779103838040, 9.539085148781746476495345714380

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