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Properties

Label 2-1805-5.4-c1-0-7
Degree $2$
Conductor $1805$
Sign $0.424 + 0.905i$
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  + 2.61i·2-s − 0.146i·3-s − 4.81·4-s + (0.948 + 2.02i)5-s + 0.383·6-s − 1.14i·7-s − 7.36i·8-s + 2.97·9-s + (−5.28 + 2.47i)10-s − 5.36·11-s + 0.707i·12-s + 2.41i·13-s + 2.97·14-s + (0.297 − 0.139i)15-s + 9.58·16-s + 5.74i·17-s + ⋯
L(s)  = 1  + 1.84i·2-s − 0.0847i·3-s − 2.40·4-s + (0.424 + 0.905i)5-s + 0.156·6-s − 0.431i·7-s − 2.60i·8-s + 0.992·9-s + (−1.67 + 0.783i)10-s − 1.61·11-s + 0.204i·12-s + 0.669i·13-s + 0.796·14-s + (0.0767 − 0.0359i)15-s + 2.39·16-s + 1.39i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.424 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.424 + 0.905i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.424 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5601135925\)
\(L(\frac12)\) \(\approx\) \(0.5601135925\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.948 - 2.02i)T \)
19 \( 1 \)
good2 \( 1 - 2.61iT - 2T^{2} \)
3 \( 1 + 0.146iT - 3T^{2} \)
7 \( 1 + 1.14iT - 7T^{2} \)
11 \( 1 + 5.36T + 11T^{2} \)
13 \( 1 - 2.41iT - 13T^{2} \)
17 \( 1 - 5.74iT - 17T^{2} \)
23 \( 1 + 1.23iT - 23T^{2} \)
29 \( 1 + 3.91T + 29T^{2} \)
31 \( 1 + 6.48T + 31T^{2} \)
37 \( 1 + 7.37iT - 37T^{2} \)
41 \( 1 + 4.86T + 41T^{2} \)
43 \( 1 - 7.59iT - 43T^{2} \)
47 \( 1 + 7.85iT - 47T^{2} \)
53 \( 1 + 0.0179iT - 53T^{2} \)
59 \( 1 + 13.0T + 59T^{2} \)
61 \( 1 + 0.0189T + 61T^{2} \)
67 \( 1 + 7.07iT - 67T^{2} \)
71 \( 1 - 2.73T + 71T^{2} \)
73 \( 1 - 5.58iT - 73T^{2} \)
79 \( 1 + 9.56T + 79T^{2} \)
83 \( 1 + 0.390iT - 83T^{2} \)
89 \( 1 + 3.57T + 89T^{2} \)
97 \( 1 + 14.7iT - 97T^{2} \)
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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

−9.772789769824525669262376426691, −8.904438079589460984053875329039, −7.925153479910566786428460583943, −7.42328978655688148939928601348, −6.87233660647874227712603920601, −6.06123696921514266811719375783, −5.39971146046724903621930159614, −4.40865833299897838058442343721, −3.58803872563836532748746902501, −1.94161431310825731272023104444, 0.20776865064613175046376301525, 1.45406198568978807795027148954, 2.40423202569007006681365612572, 3.22149581466345793064844861890, 4.39802471802305731655843086306, 5.11118689836866929112875224025, 5.57006337952069155072126904047, 7.35823834617263837237296457039, 8.175393920078201902916764701335, 9.054723623189660184010321838644

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