Properties

Label 2-975-5.4-c1-0-29
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  − 1.53i·2-s + i·3-s − 0.369·4-s + 1.53·6-s − 3.87i·7-s − 2.51i·8-s − 9-s + 1.24·11-s − 0.369i·12-s i·13-s − 5.97·14-s − 4.60·16-s − 0.659i·17-s + 1.53i·18-s − 6.97·19-s + ⋯
L(s)  = 1  − 1.08i·2-s + 0.577i·3-s − 0.184·4-s + 0.628·6-s − 1.46i·7-s − 0.887i·8-s − 0.333·9-s + 0.376·11-s − 0.106i·12-s − 0.277i·13-s − 1.59·14-s − 1.15·16-s − 0.160i·17-s + 0.362i·18-s − 1.59·19-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

\(L(1)\) \(\approx\) \(0.325766 - 1.37996i\)
\(L(\frac12)\) \(\approx\) \(0.325766 - 1.37996i\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.53iT - 2T^{2} \)
7 \( 1 + 3.87iT - 7T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
17 \( 1 + 0.659iT - 17T^{2} \)
19 \( 1 + 6.97T + 19T^{2} \)
23 \( 1 - 1.55iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 5.43T + 31T^{2} \)
37 \( 1 + 2.29iT - 37T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 + 8.20iT - 43T^{2} \)
47 \( 1 + 1.53iT - 47T^{2} \)
53 \( 1 + 1.44iT - 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 + 9.92iT - 67T^{2} \)
71 \( 1 - 11.6T + 71T^{2} \)
73 \( 1 + 6.20iT - 73T^{2} \)
79 \( 1 - 0.474T + 79T^{2} \)
83 \( 1 - 13.4iT - 83T^{2} \)
89 \( 1 - 15.9T + 89T^{2} \)
97 \( 1 + 11.6iT - 97T^{2} \)
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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

−10.15935129583940882396361596796, −9.117278863418707994112089885648, −8.136596755868652415993602835105, −7.01653609203166301330520580924, −6.36981558845086305472950148271, −4.82692553837020464877217494811, −4.00662912223271509227491943278, −3.36484439910107361669283994012, −2.03524288369887256779543882800, −0.62893996857189094187064835328, 1.91060319002650878558321971596, 2.79303442209984698353373126784, 4.53785260440403047729372683806, 5.49338314734323632599618403034, 6.45783410584167674913199245098, 6.57191462534130008080500839318, 7.930996306495751394050884997269, 8.505529903474084940452187192135, 9.039227668272106499854948977192, 10.29796123833495086515785606610

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