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Properties

Label 2-1900-95.37-c1-0-12
Degree $2$
Conductor $1900$
Sign $0.534 + 0.844i$
Analytic cond. $15.1715$
Root an. cond. $3.89507$
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  + (−1.05 − 1.05i)3-s + (−1.45 − 1.45i)7-s − 0.769i·9-s + 4.10·11-s + (3.51 + 3.51i)13-s + (−1.45 − 1.45i)17-s + (0.485 + 4.33i)19-s + 3.07i·21-s + (2.62 − 2.62i)23-s + (−3.98 + 3.98i)27-s + 10.7·29-s + 8.61i·31-s + (−4.33 − 4.33i)33-s + (−3.98 + 3.98i)37-s − 7.43i·39-s + ⋯
L(s)  = 1  + (−0.609 − 0.609i)3-s + (−0.549 − 0.549i)7-s − 0.256i·9-s + 1.23·11-s + (0.975 + 0.975i)13-s + (−0.352 − 0.352i)17-s + (0.111 + 0.993i)19-s + 0.670i·21-s + (0.547 − 0.547i)23-s + (−0.766 + 0.766i)27-s + 1.99·29-s + 1.54i·31-s + (−0.753 − 0.753i)33-s + (−0.654 + 0.654i)37-s − 1.19i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.534 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.534 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $0.534 + 0.844i$
Analytic conductor: \(15.1715\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (1557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :1/2),\ 0.534 + 0.844i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.451452289\)
\(L(\frac12)\) \(\approx\) \(1.451452289\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
19 \( 1 + (-0.485 - 4.33i)T \)
good3 \( 1 + (1.05 + 1.05i)T + 3iT^{2} \)
7 \( 1 + (1.45 + 1.45i)T + 7iT^{2} \)
11 \( 1 - 4.10T + 11T^{2} \)
13 \( 1 + (-3.51 - 3.51i)T + 13iT^{2} \)
17 \( 1 + (1.45 + 1.45i)T + 17iT^{2} \)
23 \( 1 + (-2.62 + 2.62i)T - 23iT^{2} \)
29 \( 1 - 10.7T + 29T^{2} \)
31 \( 1 - 8.61iT - 31T^{2} \)
37 \( 1 + (3.98 - 3.98i)T - 37iT^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (-1.22 + 1.22i)T - 43iT^{2} \)
47 \( 1 + (8.21 + 8.21i)T + 47iT^{2} \)
53 \( 1 + (-6.22 - 6.22i)T + 53iT^{2} \)
59 \( 1 + 5.17T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + (-5.16 + 5.16i)T - 67iT^{2} \)
71 \( 1 - 3.93iT - 71T^{2} \)
73 \( 1 + (-8.64 + 8.64i)T - 73iT^{2} \)
79 \( 1 - 3.55T + 79T^{2} \)
83 \( 1 + (-4.31 + 4.31i)T - 83iT^{2} \)
89 \( 1 + 3.07T + 89T^{2} \)
97 \( 1 + (-1.40 + 1.40i)T - 97iT^{2} \)
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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

−8.922273988066816667512939459589, −8.523773399351370385756423654834, −7.06982256347990713284911059161, −6.67170577857471361051681349225, −6.29407865015889885271128706996, −5.10070621941629191298236883476, −3.99010953342143137656936085687, −3.37336034253066280963645784220, −1.69462240309013351297121159805, −0.800851893538533654767466181019, 0.974214842709183581668809260454, 2.55652329173913937246811694107, 3.57675399845417952152452791909, 4.48147682689342093871367102616, 5.32316780187012507609993758289, 6.18391491976050947818727823914, 6.63905974244260547693058767689, 7.921130658873558043440928942239, 8.638806677839130855345965734864, 9.472432346929403964074434203465

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