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Properties

Label 2-1800-40.29-c1-0-46
Degree $2$
Conductor $1800$
Sign $0.316 - 0.948i$
Analytic cond. $14.3730$
Root an. cond. $3.79118$
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  + (1.36 + 0.366i)2-s + (1.73 + i)4-s + 2.73i·7-s + (1.99 + 2i)8-s − 2i·11-s + 3.46·13-s + (−1 + 3.73i)14-s + (1.99 + 3.46i)16-s − 3.46i·17-s + 7.46i·19-s + (0.732 − 2.73i)22-s − 4.19i·23-s + (4.73 + 1.26i)26-s + (−2.73 + 4.73i)28-s + 6.92i·29-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (0.866 + 0.5i)4-s + 1.03i·7-s + (0.707 + 0.707i)8-s − 0.603i·11-s + 0.960·13-s + (−0.267 + 0.997i)14-s + (0.499 + 0.866i)16-s − 0.840i·17-s + 1.71i·19-s + (0.156 − 0.582i)22-s − 0.874i·23-s + (0.928 + 0.248i)26-s + (−0.516 + 0.894i)28-s + 1.28i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.316 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.316 - 0.948i$
Analytic conductor: \(14.3730\)
Root analytic conductor: \(3.79118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1800,\ (\ :1/2),\ 0.316 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.409822493\)
\(L(\frac12)\) \(\approx\) \(3.409822493\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 - 0.366i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 2.73iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 3.46T + 13T^{2} \)
17 \( 1 + 3.46iT - 17T^{2} \)
19 \( 1 - 7.46iT - 19T^{2} \)
23 \( 1 + 4.19iT - 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 5.46T + 41T^{2} \)
43 \( 1 + 8.73T + 43T^{2} \)
47 \( 1 - 6.73iT - 47T^{2} \)
53 \( 1 + 4.53T + 53T^{2} \)
59 \( 1 - 0.535iT - 59T^{2} \)
61 \( 1 + 4.92iT - 61T^{2} \)
67 \( 1 + 7.26T + 67T^{2} \)
71 \( 1 - 1.46T + 71T^{2} \)
73 \( 1 - 0.535iT - 73T^{2} \)
79 \( 1 - 14.9T + 79T^{2} \)
83 \( 1 + 4.73T + 83T^{2} \)
89 \( 1 + 4.92T + 89T^{2} \)
97 \( 1 + 6.39iT - 97T^{2} \)
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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

−9.269920049544771434439734225346, −8.439449193397229794776541489819, −7.917361328644351897387616485003, −6.77096920089479665311766510064, −6.04001871966838415054976521030, −5.52333952273113681477491645753, −4.57003982773389619526367057303, −3.51061855089185071054114982131, −2.79837211638562991581086376321, −1.57985149051506913254164038015, 0.983054897969617467695536149421, 2.16408368685257094003247646809, 3.37619777175556280002289084970, 4.13382248707714998064994717799, 4.80934586604322857548505472528, 5.87446975762017910099718519072, 6.65058792096698055138610119179, 7.30811145284731836017128824824, 8.145722117640133491435350013057, 9.316523882724054254818095207223

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