Properties

Label 2-1805-5.4-c1-0-118
Degree $2$
Conductor $1805$
Sign $-0.983 + 0.182i$
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  − 2.45i·2-s + 1.56i·3-s − 4.03·4-s + (2.19 − 0.407i)5-s + 3.83·6-s − 4.50i·7-s + 4.99i·8-s + 0.563·9-s + (−1 − 5.40i)10-s + 2.19·11-s − 6.29i·12-s − 3.75i·13-s − 11.0·14-s + (0.635 + 3.43i)15-s + 4.19·16-s − 0.665i·17-s + ⋯
L(s)  = 1  − 1.73i·2-s + 0.901i·3-s − 2.01·4-s + (0.983 − 0.182i)5-s + 1.56·6-s − 1.70i·7-s + 1.76i·8-s + 0.187·9-s + (−0.316 − 1.70i)10-s + 0.662·11-s − 1.81i·12-s − 1.04i·13-s − 2.95·14-s + (0.164 + 0.886i)15-s + 1.04·16-s − 0.161i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.983 + 0.182i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.983 + 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.902338285\)
\(L(\frac12)\) \(\approx\) \(1.902338285\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.19 + 0.407i)T \)
19 \( 1 \)
good2 \( 1 + 2.45iT - 2T^{2} \)
3 \( 1 - 1.56iT - 3T^{2} \)
7 \( 1 + 4.50iT - 7T^{2} \)
11 \( 1 - 2.19T + 11T^{2} \)
13 \( 1 + 3.75iT - 13T^{2} \)
17 \( 1 + 0.665iT - 17T^{2} \)
23 \( 1 + 0.488iT - 23T^{2} \)
29 \( 1 + 3.59T + 29T^{2} \)
31 \( 1 - 6.83T + 31T^{2} \)
37 \( 1 - 3.01iT - 37T^{2} \)
41 \( 1 - 0.0724T + 41T^{2} \)
43 \( 1 - 0.420iT - 43T^{2} \)
47 \( 1 + 5.02iT - 47T^{2} \)
53 \( 1 - 2.61iT - 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 7.06T + 61T^{2} \)
67 \( 1 + 5.72iT - 67T^{2} \)
71 \( 1 - 6.97T + 71T^{2} \)
73 \( 1 + 2.95iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 15.6iT - 83T^{2} \)
89 \( 1 - 1.33T + 89T^{2} \)
97 \( 1 + 4.38iT - 97T^{2} \)
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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

−9.465172832393953573174347465085, −8.595231661229073478398877986270, −7.44783249705816940480444050936, −6.38237934051852510791521689550, −5.08200996611406346461740704427, −4.47509036754852416689373231080, −3.74107137521567807169881598465, −2.99371694333522433594829729451, −1.61551352178226160454637382829, −0.76930858935707500919458815039, 1.57972610814576505546006099158, 2.53173245670575417876754799765, 4.25324810995822957970619935575, 5.26813141298102754089395185362, 5.95022452421639469327968403691, 6.49629528574991758505944069674, 6.95884128388815762836321557968, 7.967009653954629074731858803726, 8.703239351756146430249214680814, 9.297205106258142745452560872701

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