Properties

Label 2-150-75.17-c1-0-3
Degree $2$
Conductor $150$
Sign $0.986 - 0.163i$
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  + (−0.891 − 0.453i)2-s + (0.983 + 1.42i)3-s + (0.587 + 0.809i)4-s + (1.04 − 1.97i)5-s + (−0.228 − 1.71i)6-s + (0.462 + 0.462i)7-s + (−0.156 − 0.987i)8-s + (−1.06 + 2.80i)9-s + (−1.82 + 1.28i)10-s + (2.73 − 0.888i)11-s + (−0.575 + 1.63i)12-s + (1.97 + 3.86i)13-s + (−0.202 − 0.621i)14-s + (3.84 − 0.451i)15-s + (−0.309 + 0.951i)16-s + (−5.75 + 0.911i)17-s + ⋯
L(s)  = 1  + (−0.630 − 0.321i)2-s + (0.567 + 0.823i)3-s + (0.293 + 0.404i)4-s + (0.467 − 0.883i)5-s + (−0.0932 − 0.700i)6-s + (0.174 + 0.174i)7-s + (−0.0553 − 0.349i)8-s + (−0.355 + 0.934i)9-s + (−0.578 + 0.406i)10-s + (0.824 − 0.267i)11-s + (−0.166 + 0.471i)12-s + (0.546 + 1.07i)13-s + (−0.0539 − 0.166i)14-s + (0.993 − 0.116i)15-s + (−0.0772 + 0.237i)16-s + (−1.39 + 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(150\)    =    \(2 \cdot 3 \cdot 5^{2}\)
Sign: $0.986 - 0.163i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{150} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 150,\ (\ :1/2),\ 0.986 - 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07949 + 0.0887290i\)
\(L(\frac12)\) \(\approx\) \(1.07949 + 0.0887290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.891 + 0.453i)T \)
3 \( 1 + (-0.983 - 1.42i)T \)
5 \( 1 + (-1.04 + 1.97i)T \)
good7 \( 1 + (-0.462 - 0.462i)T + 7iT^{2} \)
11 \( 1 + (-2.73 + 0.888i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.97 - 3.86i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (5.75 - 0.911i)T + (16.1 - 5.25i)T^{2} \)
19 \( 1 + (-4.28 + 5.89i)T + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.316 - 0.622i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (2.60 - 1.89i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.82 + 3.50i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.93 - 0.988i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (6.34 + 2.06i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (5.08 - 5.08i)T - 43iT^{2} \)
47 \( 1 + (-0.474 + 2.99i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (2.23 + 0.353i)T + (50.4 + 16.3i)T^{2} \)
59 \( 1 + (-2.66 + 8.19i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.77 - 8.55i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (2.22 + 14.0i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (-7.15 - 9.84i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.498 - 0.254i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (-6.00 - 8.25i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-0.742 - 4.68i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-1.78 - 5.49i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-5.26 - 0.833i)T + (92.2 + 29.9i)T^{2} \)
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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

−13.23143989941045319916132024586, −11.67707618938336797179694050763, −11.04222032330474261786185648316, −9.573725193808794318852609604929, −9.089490135494493837465201672126, −8.406819397621946007793338632245, −6.73461513981279225442306887817, −5.05902742651480412929057198393, −3.83031703548466779736714558086, −1.96710785256922718132988081386, 1.75720838939346191355360610197, 3.39675268561361150040740909875, 5.82677809182567896420708950392, 6.79502221983383947794434463393, 7.64980210951891826479348733876, 8.752883581796873580993637942441, 9.780535066758267711840594318982, 10.85490822525000572950599761846, 11.92060862362845050091378348934, 13.21678876168860398403964303103

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