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Properties

Label 2-1805-5.4-c1-0-101
Degree $2$
Conductor $1805$
Sign $0.839 - 0.543i$
Analytic cond. $14.4129$
Root an. cond. $3.79644$
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  + 0.449i·2-s + 1.95i·3-s + 1.79·4-s + (1.87 − 1.21i)5-s − 0.879·6-s − 2.06i·7-s + 1.70i·8-s − 0.829·9-s + (0.545 + 0.843i)10-s − 2.30·11-s + 3.51i·12-s − 4.39i·13-s + 0.926·14-s + (2.37 + 3.67i)15-s + 2.82·16-s − 5.47i·17-s + ⋯
L(s)  = 1  + 0.317i·2-s + 1.12i·3-s + 0.899·4-s + (0.839 − 0.543i)5-s − 0.358·6-s − 0.778i·7-s + 0.603i·8-s − 0.276·9-s + (0.172 + 0.266i)10-s − 0.695·11-s + 1.01i·12-s − 1.21i·13-s + 0.247·14-s + (0.613 + 0.948i)15-s + 0.707·16-s − 1.32i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(14.4129\)
Root analytic conductor: \(3.79644\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1805} (1084, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1805,\ (\ :1/2),\ 0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.630069753\)
\(L(\frac12)\) \(\approx\) \(2.630069753\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.87 + 1.21i)T \)
19 \( 1 \)
good2 \( 1 - 0.449iT - 2T^{2} \)
3 \( 1 - 1.95iT - 3T^{2} \)
7 \( 1 + 2.06iT - 7T^{2} \)
11 \( 1 + 2.30T + 11T^{2} \)
13 \( 1 + 4.39iT - 13T^{2} \)
17 \( 1 + 5.47iT - 17T^{2} \)
23 \( 1 - 5.81iT - 23T^{2} \)
29 \( 1 - 5.50T + 29T^{2} \)
31 \( 1 + 0.757T + 31T^{2} \)
37 \( 1 - 6.22iT - 37T^{2} \)
41 \( 1 - 6.53T + 41T^{2} \)
43 \( 1 + 3.16iT - 43T^{2} \)
47 \( 1 - 6.36iT - 47T^{2} \)
53 \( 1 + 3.85iT - 53T^{2} \)
59 \( 1 - 2.55T + 59T^{2} \)
61 \( 1 - 9.94T + 61T^{2} \)
67 \( 1 + 1.70iT - 67T^{2} \)
71 \( 1 + 9.85T + 71T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 - 2.41T + 79T^{2} \)
83 \( 1 - 7.06iT - 83T^{2} \)
89 \( 1 + 2.33T + 89T^{2} \)
97 \( 1 + 6.81iT - 97T^{2} \)
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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

−9.628198015440086909496145299424, −8.574233933266098467875372118539, −7.68576148846580952938820170203, −7.05537372125825525813073410959, −5.92774632081883711542355952488, −5.25772691787317753698416798715, −4.66890752998936663231482151664, −3.37566287188269731330912884920, −2.56466478113519372989509832854, −1.07334531821126731106111073758, 1.34770632930046797494439591199, 2.30388385068419052047526844303, 2.53615539161218854254440294910, 4.04459984865588512970459596904, 5.51569922003310481929991357822, 6.30201534916654976405125666447, 6.65011012284975674898443096056, 7.44213799053998538037632301737, 8.321988744296589455923967172773, 9.176661138551404341515304369136

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