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Properties

Label 2-150-75.8-c1-0-7
Degree $2$
Conductor $150$
Sign $0.408 + 0.912i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
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  + (0.453 − 0.891i)2-s + (1.60 − 0.646i)3-s + (−0.587 − 0.809i)4-s + (−1.92 + 1.14i)5-s + (0.153 − 1.72i)6-s + (2.03 − 2.03i)7-s + (−0.987 + 0.156i)8-s + (2.16 − 2.07i)9-s + (0.148 + 2.23i)10-s + (2.60 − 0.847i)11-s + (−1.46 − 0.920i)12-s + (−5.27 + 2.68i)13-s + (−0.891 − 2.74i)14-s + (−2.34 + 3.08i)15-s + (−0.309 + 0.951i)16-s + (0.936 + 5.91i)17-s + ⋯
L(s)  = 1  + (0.321 − 0.630i)2-s + (0.927 − 0.373i)3-s + (−0.293 − 0.404i)4-s + (−0.858 + 0.512i)5-s + (0.0627 − 0.704i)6-s + (0.770 − 0.770i)7-s + (−0.349 + 0.0553i)8-s + (0.721 − 0.692i)9-s + (0.0468 + 0.705i)10-s + (0.786 − 0.255i)11-s + (−0.423 − 0.265i)12-s + (−1.46 + 0.744i)13-s + (−0.238 − 0.733i)14-s + (−0.605 + 0.795i)15-s + (−0.0772 + 0.237i)16-s + (0.227 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.408 + 0.912i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.408 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28081 - 0.829923i\)
\(L(\frac12)\) \(\approx\) \(1.28081 - 0.829923i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.453 + 0.891i)T \)
3 \( 1 + (-1.60 + 0.646i)T \)
5 \( 1 + (1.92 - 1.14i)T \)
good7 \( 1 + (-2.03 + 2.03i)T - 7iT^{2} \)
11 \( 1 + (-2.60 + 0.847i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (5.27 - 2.68i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.936 - 5.91i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (3.04 - 4.19i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.01 + 0.515i)T + (13.5 + 18.6i)T^{2} \)
29 \( 1 + (2.34 - 1.70i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.56 - 3.31i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.66 + 7.18i)T + (-21.7 + 29.9i)T^{2} \)
41 \( 1 + (-5.02 - 1.63i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (3.10 + 3.10i)T + 43iT^{2} \)
47 \( 1 + (4.58 + 0.726i)T + (44.6 + 14.5i)T^{2} \)
53 \( 1 + (-0.837 + 5.29i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-0.912 + 2.80i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.41 + 4.36i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (0.455 - 0.0721i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (3.36 + 4.62i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.12 + 8.09i)T + (-42.9 - 59.0i)T^{2} \)
79 \( 1 + (-6.90 - 9.50i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.9 - 1.88i)T + (78.9 - 25.6i)T^{2} \)
89 \( 1 + (0.402 + 1.23i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-1.03 + 6.52i)T + (-92.2 - 29.9i)T^{2} \)
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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

−12.63976436117135727717871038243, −12.02104260824703330901933561861, −10.88273378089849123361911571822, −9.940960544989349823458644187227, −8.571760556910565799442730492605, −7.66882415899510790470625799131, −6.60520878951556301131006767025, −4.38298635350630020132723515402, −3.62097810395733975301993653314, −1.86220686887543127490322511151, 2.75033933778014118970254325193, 4.43183337248048599686216735078, 5.10177095676143175013805341854, 7.15581937683713187205994528534, 7.970918116394997124927928526414, 8.864729340975775416955006642429, 9.747904554790494962502634802133, 11.51772249846179439797558568717, 12.26080764974900492318818143563, 13.35543186146610990941926913467

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