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Properties

Label 2-675-15.2-c1-0-22
Degree $2$
Conductor $675$
Sign $-0.973 + 0.229i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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  + (1.79 − 1.79i)2-s − 4.44i·4-s + (−2.22 − 2.22i)7-s + (−4.39 − 4.39i)8-s − 0.807i·11-s + (0.775 − 0.775i)13-s − 7.99·14-s − 6.89·16-s + (0.807 − 0.807i)17-s + 3i·19-s + (−1.44 − 1.44i)22-s + (−1.79 − 1.79i)23-s − 2.78i·26-s + (−9.89 + 9.89i)28-s + 6.37·29-s + ⋯
L(s)  = 1  + (1.26 − 1.26i)2-s − 2.22i·4-s + (−0.840 − 0.840i)7-s + (−1.55 − 1.55i)8-s − 0.243i·11-s + (0.215 − 0.215i)13-s − 2.13·14-s − 1.72·16-s + (0.195 − 0.195i)17-s + 0.688i·19-s + (−0.309 − 0.309i)22-s + (−0.374 − 0.374i)23-s − 0.546i·26-s + (−1.87 + 1.87i)28-s + 1.18·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.289100 - 2.48295i\)
\(L(\frac12)\) \(\approx\) \(0.289100 - 2.48295i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-1.79 + 1.79i)T - 2iT^{2} \)
7 \( 1 + (2.22 + 2.22i)T + 7iT^{2} \)
11 \( 1 + 0.807iT - 11T^{2} \)
13 \( 1 + (-0.775 + 0.775i)T - 13iT^{2} \)
17 \( 1 + (-0.807 + 0.807i)T - 17iT^{2} \)
19 \( 1 - 3iT - 19T^{2} \)
23 \( 1 + (1.79 + 1.79i)T + 23iT^{2} \)
29 \( 1 - 6.37T + 29T^{2} \)
31 \( 1 + 5.34T + 31T^{2} \)
37 \( 1 + (-5.67 - 5.67i)T + 37iT^{2} \)
41 \( 1 + 11.5iT - 41T^{2} \)
43 \( 1 + (3.44 - 3.44i)T - 43iT^{2} \)
47 \( 1 + (-7.18 + 7.18i)T - 47iT^{2} \)
53 \( 1 + (-7.00 - 7.00i)T + 53iT^{2} \)
59 \( 1 - 6.82T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + (-6.22 - 6.22i)T + 67iT^{2} \)
71 \( 1 - 9.96iT - 71T^{2} \)
73 \( 1 + (-2.67 + 2.67i)T - 73iT^{2} \)
79 \( 1 + 0.550iT - 79T^{2} \)
83 \( 1 + (-2.60 - 2.60i)T + 83iT^{2} \)
89 \( 1 + 10.7T + 89T^{2} \)
97 \( 1 + (-2.12 - 2.12i)T + 97iT^{2} \)
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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

−10.26003173652314422371245353755, −9.860631904652056018918972705046, −8.562475451207104797606328693343, −7.17064101647704106549487361093, −6.17950620266781329312243364739, −5.33654199054641800946785172404, −4.13409198033614155900197758655, −3.54152613561340717493771787183, −2.46582085224903744607586868241, −0.914050157875141124082925466843, 2.60967387060354627877961696769, 3.63769503207605288516546458393, 4.67014416964302427701792958459, 5.64222411367099647747299328030, 6.30182741400781134186629869948, 7.06498506321954841136708042884, 8.025534355677522228351088127906, 8.925500436722689537511186853082, 9.806876410916978165347869428487, 11.21459155166812150201012779394

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