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Properties

Label 2-30e2-9.4-c1-0-8
Degree $2$
Conductor $900$
Sign $0.688 - 0.725i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + (1.72 − 0.0977i)3-s + (1.70 + 2.94i)7-s + (2.98 − 0.338i)9-s + (2.20 + 3.81i)11-s + (−1.03 + 1.78i)13-s − 1.40·17-s − 6.35·19-s + (3.22 + 4.92i)21-s + (−0.0539 + 0.0933i)23-s + (5.12 − 0.875i)27-s + (−4.54 − 7.86i)29-s + (−1.53 + 2.65i)31-s + (4.17 + 6.37i)33-s + 1.95·37-s + (−1.60 + 3.19i)39-s + ⋯
L(s)  = 1  + (0.998 − 0.0564i)3-s + (0.642 + 1.11i)7-s + (0.993 − 0.112i)9-s + (0.663 + 1.14i)11-s + (−0.286 + 0.495i)13-s − 0.339·17-s − 1.45·19-s + (0.704 + 1.07i)21-s + (−0.0112 + 0.0194i)23-s + (0.985 − 0.168i)27-s + (−0.843 − 1.46i)29-s + (−0.275 + 0.476i)31-s + (0.727 + 1.11i)33-s + 0.321·37-s + (−0.257 + 0.510i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.688 - 0.725i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.688 - 0.725i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17001 + 0.931636i\)
\(L(\frac12)\) \(\approx\) \(2.17001 + 0.931636i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.72 + 0.0977i)T \)
5 \( 1 \)
good7 \( 1 + (-1.70 - 2.94i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.20 - 3.81i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.03 - 1.78i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.40T + 17T^{2} \)
19 \( 1 + 6.35T + 19T^{2} \)
23 \( 1 + (0.0539 - 0.0933i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.54 + 7.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.53 - 2.65i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.95T + 37T^{2} \)
41 \( 1 + (-4.34 + 7.53i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.56 - 6.17i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.74 - 6.48i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 13.0T + 53T^{2} \)
59 \( 1 + (-6.58 + 11.4i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.862 - 1.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.18 + 10.7i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.50T + 71T^{2} \)
73 \( 1 + 5.42T + 73T^{2} \)
79 \( 1 + (4.71 + 8.17i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.48 + 7.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.01T + 89T^{2} \)
97 \( 1 + (1.08 + 1.88i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.952540684890777052110737600871, −9.191483537558264565598330770099, −8.706739306686151092997903687042, −7.77920371819936029362508460040, −6.96533555185658937282144150603, −5.96062460593936548825033637460, −4.62559444187711526982301823300, −4.02024352939577709614007182210, −2.37925802480055375802088135283, −1.92070818782146938998829385616, 1.09526816492604661025867385443, 2.46699257661948749758894440323, 3.77081684847438041048593449940, 4.26351004061246734984516989670, 5.60629014198186742927884258844, 6.87013902544806260828809071909, 7.49220006956589703433714707979, 8.514876747478092565534430796997, 8.861979060050396037637076306685, 10.07316187022479218087085318973

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