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Properties

Label 2-150-5.4-c3-0-4
Degree $2$
Conductor $150$
Sign $0.894 - 0.447i$
Analytic cond. $8.85028$
Root an. cond. $2.97494$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2i·2-s + 3i·3-s − 4·4-s + 6·6-s − i·7-s + 8i·8-s − 9·9-s + 42·11-s − 12i·12-s + 67i·13-s − 2·14-s + 16·16-s + 54i·17-s + 18i·18-s + 115·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.0539i·7-s + 0.353i·8-s − 0.333·9-s + 1.15·11-s − 0.288i·12-s + 1.42i·13-s − 0.0381·14-s + 0.250·16-s + 0.770i·17-s + 0.235i·18-s + 1.38·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(8.85028\)
Root analytic conductor: \(2.97494\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :3/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.51767 + 0.358273i\)
\(L(\frac12)\) \(\approx\) \(1.51767 + 0.358273i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
3 \( 1 - 3iT \)
5 \( 1 \)
good7 \( 1 + iT - 343T^{2} \)
11 \( 1 - 42T + 1.33e3T^{2} \)
13 \( 1 - 67iT - 2.19e3T^{2} \)
17 \( 1 - 54iT - 4.91e3T^{2} \)
19 \( 1 - 115T + 6.85e3T^{2} \)
23 \( 1 - 162iT - 1.21e4T^{2} \)
29 \( 1 - 210T + 2.43e4T^{2} \)
31 \( 1 + 193T + 2.97e4T^{2} \)
37 \( 1 + 286iT - 5.06e4T^{2} \)
41 \( 1 - 12T + 6.89e4T^{2} \)
43 \( 1 + 263iT - 7.95e4T^{2} \)
47 \( 1 - 414iT - 1.03e5T^{2} \)
53 \( 1 - 192iT - 1.48e5T^{2} \)
59 \( 1 + 690T + 2.05e5T^{2} \)
61 \( 1 + 733T + 2.26e5T^{2} \)
67 \( 1 - 299iT - 3.00e5T^{2} \)
71 \( 1 + 228T + 3.57e5T^{2} \)
73 \( 1 + 938iT - 3.89e5T^{2} \)
79 \( 1 - 160T + 4.93e5T^{2} \)
83 \( 1 - 462iT - 5.71e5T^{2} \)
89 \( 1 - 240T + 7.04e5T^{2} \)
97 \( 1 + 511iT - 9.12e5T^{2} \)
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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

−12.27926975492936258042810715495, −11.61708968336869741702592152970, −10.69756348222730469370161281184, −9.425627480780673480656399868532, −9.038358464480470032167562830387, −7.38139328626076788379556180652, −5.90350951884859667764036100025, −4.43486644634636739135977775392, −3.46514760153736465967703946633, −1.55349031487267319906828889420, 0.864941365152578160586413356200, 3.12373935530626252317664362402, 4.89227719407158069772713151397, 6.11117726628626872338109923775, 7.11223521662485343167009696141, 8.119807602237743918395530865958, 9.145745286441402984217522441229, 10.31581325685522804454997161309, 11.72882092462804466760756431651, 12.54057490409254281757965282229

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