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Properties

Label 2-1805-5.4-c1-0-125
Degree $2$
Conductor $1805$
Sign $0.864 + 0.503i$
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  + 0.578i·2-s − 0.551i·3-s + 1.66·4-s + (1.93 + 1.12i)5-s + 0.319·6-s − 4.66i·7-s + 2.11i·8-s + 2.69·9-s + (−0.650 + 1.11i)10-s − 1.22·11-s − 0.919i·12-s − 5.34i·13-s + 2.69·14-s + (0.620 − 1.06i)15-s + 2.10·16-s + 1.41i·17-s + ⋯
L(s)  = 1  + 0.408i·2-s − 0.318i·3-s + 0.832·4-s + (0.864 + 0.503i)5-s + 0.130·6-s − 1.76i·7-s + 0.749i·8-s + 0.898·9-s + (−0.205 + 0.353i)10-s − 0.369·11-s − 0.265i·12-s − 1.48i·13-s + 0.720·14-s + (0.160 − 0.275i)15-s + 0.526·16-s + 0.342i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 + 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1805\)    =    \(5 \cdot 19^{2}\)
Sign: $0.864 + 0.503i$
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: \((2,\ 1805,\ (\ :1/2),\ 0.864 + 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.680911496\)
\(L(\frac12)\) \(\approx\) \(2.680911496\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.93 - 1.12i)T \)
19 \( 1 \)
good2 \( 1 - 0.578iT - 2T^{2} \)
3 \( 1 + 0.551iT - 3T^{2} \)
7 \( 1 + 4.66iT - 7T^{2} \)
11 \( 1 + 1.22T + 11T^{2} \)
13 \( 1 + 5.34iT - 13T^{2} \)
17 \( 1 - 1.41iT - 17T^{2} \)
23 \( 1 + 1.95iT - 23T^{2} \)
29 \( 1 + 7.32T + 29T^{2} \)
31 \( 1 + 1.83T + 31T^{2} \)
37 \( 1 + 5.59iT - 37T^{2} \)
41 \( 1 - 8.29T + 41T^{2} \)
43 \( 1 + 8.30iT - 43T^{2} \)
47 \( 1 - 4.10iT - 47T^{2} \)
53 \( 1 - 12.9iT - 53T^{2} \)
59 \( 1 + 3.76T + 59T^{2} \)
61 \( 1 + 4.63T + 61T^{2} \)
67 \( 1 - 4.65iT - 67T^{2} \)
71 \( 1 + 8.44T + 71T^{2} \)
73 \( 1 - 4.99iT - 73T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 - 1.02iT - 83T^{2} \)
89 \( 1 + 3.94T + 89T^{2} \)
97 \( 1 - 4.72iT - 97T^{2} \)
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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

−9.338412571220235379766878084315, −7.901910321814272906140718362565, −7.46968643963490455200983434927, −7.02557776858123744448296384287, −6.11216154020907628062223305525, −5.45957704352634275938225361175, −4.18930702471175161017433046986, −3.17834295627870042722642794698, −2.08101576329719828411181012256, −1.00451971208845258222304363303, 1.68544183079392578049080933889, 2.09803389982137216318938034902, 3.17235111132651730890723961748, 4.46171406293461845413773368914, 5.32939431124837865425872189426, 6.09269166665759305315765342598, 6.77878056020369558099347231692, 7.78931703186635733486564000525, 8.906493766463536148101191332718, 9.482295294891164103483250962712

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