Properties

Label 2-1900-380.227-c0-0-7
Degree $2$
Conductor $1900$
Sign $-0.973 + 0.229i$
Analytic cond. $0.948223$
Root an. cond. $0.973767$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 − 0.923i)2-s + (1.30 − 1.30i)3-s + (−0.707 − 0.707i)4-s + (−0.707 − 1.70i)6-s + (−0.923 + 0.382i)8-s − 2.41i·9-s + 1.41i·11-s − 1.84·12-s + (−1.30 − 1.30i)13-s + i·16-s + (−2.23 − 0.923i)18-s + 19-s + (1.30 + 0.541i)22-s + (−0.707 + 1.70i)24-s + (−1.70 + 0.707i)26-s + (−1.84 − 1.84i)27-s + ⋯
L(s)  = 1  + (0.382 − 0.923i)2-s + (1.30 − 1.30i)3-s + (−0.707 − 0.707i)4-s + (−0.707 − 1.70i)6-s + (−0.923 + 0.382i)8-s − 2.41i·9-s + 1.41i·11-s − 1.84·12-s + (−1.30 − 1.30i)13-s + i·16-s + (−2.23 − 0.923i)18-s + 19-s + (1.30 + 0.541i)22-s + (−0.707 + 1.70i)24-s + (−1.70 + 0.707i)26-s + (−1.84 − 1.84i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1900\)    =    \(2^{2} \cdot 5^{2} \cdot 19\)
Sign: $-0.973 + 0.229i$
Analytic conductor: \(0.948223\)
Root analytic conductor: \(0.973767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1900} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1900,\ (\ :0),\ -0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.832074668\)
\(L(\frac12)\) \(\approx\) \(1.832074668\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 \)
19 \( 1 - T \)
good3 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + (1.30 + 1.30i)T + iT^{2} \)
17 \( 1 + iT^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + (-0.541 - 0.541i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
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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

−9.226794020253670580631519748069, −8.188564413639549159227791267304, −7.54083111431792685847880665059, −6.97014170306568899729369432647, −5.75862247154546065618286476282, −4.80401917601031926557068542042, −3.68979287640382563508939471400, −2.71710528807948341605507902848, −2.26597779040219881941539119828, −1.09114076169499425814473045388, 2.46411117762019502850953126992, 3.32533611319583531106423311152, 4.01514808232187882780723276098, 4.85836513371519617267666041308, 5.48743641422724066387026333668, 6.71756971969650903539382221091, 7.57652977702784481390779753621, 8.322243184094183552609890507362, 8.879785385300731468245046093483, 9.570907780152411265683880719247

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