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Properties

Label 2-1805-5.4-c1-0-16
Degree $2$
Conductor $1805$
Sign $-0.999 - 0.0443i$
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  + 1.22i·2-s − 0.804i·3-s + 0.507·4-s + (−2.23 − 0.0991i)5-s + 0.982·6-s + 3.79i·7-s + 3.06i·8-s + 2.35·9-s + (0.121 − 2.72i)10-s − 1.23·11-s − 0.408i·12-s + 3.45i·13-s − 4.63·14-s + (−0.0797 + 1.79i)15-s − 2.72·16-s + 3.00i·17-s + ⋯
L(s)  = 1  + 0.863i·2-s − 0.464i·3-s + 0.253·4-s + (−0.999 − 0.0443i)5-s + 0.401·6-s + 1.43i·7-s + 1.08i·8-s + 0.784·9-s + (0.0382 − 0.862i)10-s − 0.373·11-s − 0.117i·12-s + 0.958i·13-s − 1.23·14-s + (−0.0205 + 0.463i)15-s − 0.681·16-s + 0.729i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0443i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0443i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $-0.999 - 0.0443i$
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: \((2,\ 1805,\ (\ :1/2),\ -0.999 - 0.0443i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.189815687\)
\(L(\frac12)\) \(\approx\) \(1.189815687\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.23 + 0.0991i)T \)
19 \( 1 \)
good2 \( 1 - 1.22iT - 2T^{2} \)
3 \( 1 + 0.804iT - 3T^{2} \)
7 \( 1 - 3.79iT - 7T^{2} \)
11 \( 1 + 1.23T + 11T^{2} \)
13 \( 1 - 3.45iT - 13T^{2} \)
17 \( 1 - 3.00iT - 17T^{2} \)
23 \( 1 + 6.14iT - 23T^{2} \)
29 \( 1 + 4.28T + 29T^{2} \)
31 \( 1 + 5.10T + 31T^{2} \)
37 \( 1 + 9.13iT - 37T^{2} \)
41 \( 1 + 5.33T + 41T^{2} \)
43 \( 1 - 9.12iT - 43T^{2} \)
47 \( 1 - 7.30iT - 47T^{2} \)
53 \( 1 + 3.33iT - 53T^{2} \)
59 \( 1 + 0.817T + 59T^{2} \)
61 \( 1 - 6.40T + 61T^{2} \)
67 \( 1 - 1.01iT - 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 11.0iT - 73T^{2} \)
79 \( 1 + 1.38T + 79T^{2} \)
83 \( 1 - 2.52iT - 83T^{2} \)
89 \( 1 - 2.69T + 89T^{2} \)
97 \( 1 + 3.06iT - 97T^{2} \)
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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

−9.268628796903461369169562678691, −8.613377408619180267284933552822, −7.958028499633441138436039909630, −7.27474947911620170430420169044, −6.58262860124378396989519294465, −5.87387491981648813075061634517, −4.94016240597667533267574742071, −4.00677120572285102050179425948, −2.62714022672426592092935888238, −1.78109993109591185695643448457, 0.43000810504227161654835886566, 1.57213199680127658974877007125, 3.19742538996373936021269983574, 3.62138275419632419628486399885, 4.40097258219362561788925878690, 5.35524455147041262612717117609, 6.91077099065764646551775601474, 7.29402524631420414470762446002, 7.88158994609028337312479228693, 9.145121840685712580193028077468

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