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Properties

Label 2-1805-5.4-c1-0-53
Degree $2$
Conductor $1805$
Sign $0.930 + 0.366i$
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  − 0.249i·2-s − 2.30i·3-s + 1.93·4-s + (−2.08 − 0.820i)5-s − 0.574·6-s + 3.96i·7-s − 0.983i·8-s − 2.29·9-s + (−0.204 + 0.519i)10-s + 3.13·11-s − 4.45i·12-s + 2.65i·13-s + 0.989·14-s + (−1.88 + 4.78i)15-s + 3.62·16-s + 2.25i·17-s + ⋯
L(s)  = 1  − 0.176i·2-s − 1.32i·3-s + 0.968·4-s + (−0.930 − 0.366i)5-s − 0.234·6-s + 1.49i·7-s − 0.347i·8-s − 0.765·9-s + (−0.0648 + 0.164i)10-s + 0.945·11-s − 1.28i·12-s + 0.737i·13-s + 0.264·14-s + (−0.487 + 1.23i)15-s + 0.907·16-s + 0.547i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.366i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.930 + 0.366i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.930 + 0.366i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.951090763\)
\(L(\frac12)\) \(\approx\) \(1.951090763\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.08 + 0.820i)T \)
19 \( 1 \)
good2 \( 1 + 0.249iT - 2T^{2} \)
3 \( 1 + 2.30iT - 3T^{2} \)
7 \( 1 - 3.96iT - 7T^{2} \)
11 \( 1 - 3.13T + 11T^{2} \)
13 \( 1 - 2.65iT - 13T^{2} \)
17 \( 1 - 2.25iT - 17T^{2} \)
23 \( 1 - 7.58iT - 23T^{2} \)
29 \( 1 + 1.36T + 29T^{2} \)
31 \( 1 - 0.894T + 31T^{2} \)
37 \( 1 - 6.62iT - 37T^{2} \)
41 \( 1 - 6.23T + 41T^{2} \)
43 \( 1 - 1.77iT - 43T^{2} \)
47 \( 1 - 0.176iT - 47T^{2} \)
53 \( 1 + 6.77iT - 53T^{2} \)
59 \( 1 - 7.81T + 59T^{2} \)
61 \( 1 + 1.03T + 61T^{2} \)
67 \( 1 - 15.2iT - 67T^{2} \)
71 \( 1 - 10.4T + 71T^{2} \)
73 \( 1 + 4.18iT - 73T^{2} \)
79 \( 1 + 11.2T + 79T^{2} \)
83 \( 1 + 13.2iT - 83T^{2} \)
89 \( 1 + 1.48T + 89T^{2} \)
97 \( 1 + 15.3iT - 97T^{2} \)
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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

−8.991605923323459732426101157859, −8.314293643767279246929514374485, −7.57194948006921303321399441156, −6.88086784388234776196693446246, −6.22826595716402938547114337897, −5.46764171606228076102564931193, −4.05998740756939924633654180348, −3.01840504332192120141890405289, −1.96062079973535664190712991928, −1.28711574047018909461658736975, 0.798886092778887115805444333131, 2.70142220203964984413654418897, 3.70993705183806302054874007142, 4.10404945213730999927096235430, 5.03687996880644384009021502942, 6.28360725823560979177554252147, 7.04071176551789935472541398540, 7.59245713269480077788972916520, 8.459729119698570993282849071811, 9.501366875065542593283794729101

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