Properties

Label 2-75-15.2-c9-0-29
Degree $2$
Conductor $75$
Sign $0.973 - 0.229i$
Analytic cond. $38.6276$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−99.2 + 99.2i)3-s + 512i·4-s + (1.25e3 + 1.25e3i)7-s − 1.96e4i·9-s + (−5.07e4 − 5.07e4i)12-s + (1.19e5 − 1.19e5i)13-s − 2.62e5·16-s − 9.76e5i·19-s − 2.49e5·21-s + (1.95e6 + 1.95e6i)27-s + (−6.43e5 + 6.43e5i)28-s − 1.69e6·31-s + 1.00e7·36-s + (1.18e7 + 1.18e7i)37-s + 2.36e7i·39-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s + i·4-s + (0.197 + 0.197i)7-s − 0.999i·9-s + (−0.707 − 0.707i)12-s + (1.15 − 1.15i)13-s − 16-s − 1.71i·19-s − 0.279·21-s + (0.707 + 0.707i)27-s + (−0.197 + 0.197i)28-s − 0.328·31-s + 0.999·36-s + (1.04 + 1.04i)37-s + 1.63i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{75} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ 0.973 - 0.229i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.44717 + 0.168500i\)
\(L(\frac12)\) \(\approx\) \(1.44717 + 0.168500i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (99.2 - 99.2i)T \)
5 \( 1 \)
good2 \( 1 - 512iT^{2} \)
7 \( 1 + (-1.25e3 - 1.25e3i)T + 4.03e7iT^{2} \)
11 \( 1 - 2.35e9T^{2} \)
13 \( 1 + (-1.19e5 + 1.19e5i)T - 1.06e10iT^{2} \)
17 \( 1 - 1.18e11iT^{2} \)
19 \( 1 + 9.76e5iT - 3.22e11T^{2} \)
23 \( 1 + 1.80e12iT^{2} \)
29 \( 1 + 1.45e13T^{2} \)
31 \( 1 + 1.69e6T + 2.64e13T^{2} \)
37 \( 1 + (-1.18e7 - 1.18e7i)T + 1.29e14iT^{2} \)
41 \( 1 - 3.27e14T^{2} \)
43 \( 1 + (-2.94e7 + 2.94e7i)T - 5.02e14iT^{2} \)
47 \( 1 - 1.11e15iT^{2} \)
53 \( 1 + 3.29e15iT^{2} \)
59 \( 1 + 8.66e15T^{2} \)
61 \( 1 + 1.17e8T + 1.16e16T^{2} \)
67 \( 1 + (-2.19e8 - 2.19e8i)T + 2.72e16iT^{2} \)
71 \( 1 - 4.58e16T^{2} \)
73 \( 1 + (-2.71e8 + 2.71e8i)T - 5.88e16iT^{2} \)
79 \( 1 - 6.16e8iT - 1.19e17T^{2} \)
83 \( 1 + 1.86e17iT^{2} \)
89 \( 1 + 3.50e17T^{2} \)
97 \( 1 + (8.30e8 + 8.30e8i)T + 7.60e17iT^{2} \)
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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

−12.60785456760721025235209173784, −11.47480276628796893196777756890, −10.75713975614614920685534034779, −9.260607552651802014235421339547, −8.246731677815366425356653935441, −6.78162504429410445762406286462, −5.40162171196117831722805179781, −4.10665209555466161875698577355, −2.90884596816198060266425717616, −0.59286205610586884259451132353, 0.995232416333677329534973895786, 1.87880342569820562167800176447, 4.27778403450699094046958501259, 5.74546662995937661657476214109, 6.45522241995490734489996448069, 7.84652813924331521823007899308, 9.355900528283155175617675070638, 10.69532687626409143522619213003, 11.34974522122089153277149846865, 12.58077584199314948693337000082

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