Properties

Label 2-975-5.4-c1-0-14
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + i·3-s − 2·4-s + 2·6-s + i·7-s − 9-s + 5·11-s − 2i·12-s i·13-s + 2·14-s − 4·16-s + 7i·17-s + 2i·18-s + 6·19-s − 21-s − 10i·22-s + ⋯
L(s)  = 1  − 1.41i·2-s + 0.577i·3-s − 4-s + 0.816·6-s + 0.377i·7-s − 0.333·9-s + 1.50·11-s − 0.577i·12-s − 0.277i·13-s + 0.534·14-s − 16-s + 1.69i·17-s + 0.471i·18-s + 1.37·19-s − 0.218·21-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: \(0\)
Selberg data: \((2,\ 975,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46799 - 0.907267i\)
\(L(\frac12)\) \(\approx\) \(1.46799 - 0.907267i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 7T^{2} \)
11 \( 1 - 5T + 11T^{2} \)
17 \( 1 - 7iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 + 10iT - 47T^{2} \)
53 \( 1 - 5iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 5T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 - 9T + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 3T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 11T + 89T^{2} \)
97 \( 1 - 11iT - 97T^{2} \)
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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

−9.996394008838985257822741622651, −9.243254900933984545463546409236, −8.740472855889520499932547426011, −7.42827363512632460370386271467, −6.25274741289702926900474348398, −5.32632479725822535632326487500, −3.86281532014586325850132885485, −3.73116461641501034639507186577, −2.32357140714888001353167288909, −1.16527492879914064662368930022, 1.06561585539677549750123855956, 2.79294761687251196328087093584, 4.25303219057324838174535968171, 5.16828518296412863150588415303, 6.18490496518153754585792650511, 6.84709493670711635863691803818, 7.39288382163713073087305494813, 8.187482288672853866644552945813, 9.217628409188435316700700330816, 9.633176476545315607029227176635

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