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Properties

Label 2-1805-5.4-c1-0-58
Degree $2$
Conductor $1805$
Sign $0.545 - 0.838i$
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  − 0.113i·2-s + 1.07i·3-s + 1.98·4-s + (−1.21 + 1.87i)5-s + 0.122·6-s − 1.50i·7-s − 0.452i·8-s + 1.84·9-s + (0.212 + 0.138i)10-s − 0.314·11-s + 2.13i·12-s − 5.17i·13-s − 0.170·14-s + (−2.01 − 1.31i)15-s + 3.92·16-s + 6.53i·17-s + ⋯
L(s)  = 1  − 0.0802i·2-s + 0.621i·3-s + 0.993·4-s + (−0.545 + 0.838i)5-s + 0.0498·6-s − 0.569i·7-s − 0.159i·8-s + 0.613·9-s + (0.0672 + 0.0437i)10-s − 0.0948·11-s + 0.617i·12-s − 1.43i·13-s − 0.0456·14-s + (−0.520 − 0.338i)15-s + 0.980·16-s + 1.58i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.545 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.545 - 0.838i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.545 - 0.838i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.190638931\)
\(L(\frac12)\) \(\approx\) \(2.190638931\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.21 - 1.87i)T \)
19 \( 1 \)
good2 \( 1 + 0.113iT - 2T^{2} \)
3 \( 1 - 1.07iT - 3T^{2} \)
7 \( 1 + 1.50iT - 7T^{2} \)
11 \( 1 + 0.314T + 11T^{2} \)
13 \( 1 + 5.17iT - 13T^{2} \)
17 \( 1 - 6.53iT - 17T^{2} \)
23 \( 1 - 5.96iT - 23T^{2} \)
29 \( 1 - 6.64T + 29T^{2} \)
31 \( 1 - 7.01T + 31T^{2} \)
37 \( 1 + 1.92iT - 37T^{2} \)
41 \( 1 + 1.75T + 41T^{2} \)
43 \( 1 - 0.0943iT - 43T^{2} \)
47 \( 1 - 0.794iT - 47T^{2} \)
53 \( 1 + 6.17iT - 53T^{2} \)
59 \( 1 + 12.6T + 59T^{2} \)
61 \( 1 - 7.61T + 61T^{2} \)
67 \( 1 - 14.5iT - 67T^{2} \)
71 \( 1 + 2.29T + 71T^{2} \)
73 \( 1 - 8.53iT - 73T^{2} \)
79 \( 1 - 8.30T + 79T^{2} \)
83 \( 1 - 3.71iT - 83T^{2} \)
89 \( 1 + 2.77T + 89T^{2} \)
97 \( 1 + 6.44iT - 97T^{2} \)
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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

−9.862711914128937437122750332820, −8.316289394553872216517956572759, −7.79811774173287587884008251454, −7.04388008510138125841856368315, −6.33124954692120463718676106287, −5.41873300231242135439961371094, −4.14195343434905451374892782330, −3.51499425067908554487155391315, −2.68618155124951986362866575442, −1.25103918965834264979157188643, 0.928924828479740409286850383857, 2.02233830294837514833751424680, 2.90371664563810494754680993961, 4.37491275469894498720267911857, 4.95382187084229365769432254116, 6.26333286551371055294801689227, 6.76362867962750995087222110884, 7.50474279565739599292852562348, 8.234733026363926914729890049450, 9.033508995932345786999543746896

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