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Properties

Label 2-945-945.209-c1-0-32
Degree $2$
Conductor $945$
Sign $-0.720 - 0.693i$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
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  + (−1.62 + 0.586i)3-s + (−1.53 + 1.28i)4-s + (−2.20 + 0.388i)5-s + (2.02 + 1.70i)7-s + (2.31 − 1.91i)9-s + (4.46 + 0.787i)11-s + (1.74 − 2.99i)12-s + (6.63 + 2.41i)13-s + (3.36 − 1.92i)15-s + (0.694 − 3.93i)16-s + (−5.90 + 3.40i)17-s + (2.87 − 3.42i)20-s + (−4.30 − 1.58i)21-s + (4.69 − 1.71i)25-s + (−2.64 + 4.47i)27-s − 5.29·28-s + ⋯
L(s)  = 1  + (−0.940 + 0.338i)3-s + (−0.766 + 0.642i)4-s + (−0.984 + 0.173i)5-s + (0.766 + 0.642i)7-s + (0.770 − 0.637i)9-s + (1.34 + 0.237i)11-s + (0.503 − 0.864i)12-s + (1.84 + 0.670i)13-s + (0.867 − 0.496i)15-s + (0.173 − 0.984i)16-s + (−1.43 + 0.826i)17-s + (0.642 − 0.766i)20-s + (−0.938 − 0.345i)21-s + (0.939 − 0.342i)25-s + (−0.509 + 0.860i)27-s − 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 - 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.720 - 0.693i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.720 - 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.309793 + 0.767879i\)
\(L(\frac12)\) \(\approx\) \(0.309793 + 0.767879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.62 - 0.586i)T \)
5 \( 1 + (2.20 - 0.388i)T \)
7 \( 1 + (-2.02 - 1.70i)T \)
good2 \( 1 + (1.53 - 1.28i)T^{2} \)
11 \( 1 + (-4.46 - 0.787i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (-6.63 - 2.41i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (5.90 - 3.40i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-1.65 - 4.54i)T + (-22.2 + 18.6i)T^{2} \)
31 \( 1 + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (6.27 - 7.47i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-1.16 + 0.675i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-8.36 + 14.4i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.58 - 2.39i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (4.73 + 12.9i)T + (-63.5 + 53.3i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.71 - 9.72i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.70284514962334853017794117552, −9.154077617495231191082100398696, −8.853269551049331307134363525854, −8.027713185658861499165987225189, −6.79243931699036435220363869021, −6.18245041916422969678596517580, −4.79386723930106604218512251496, −4.19237276058825182958677647191, −3.56573918874716103678491628963, −1.39103761686167038912397822960, 0.56649515709997580358833206322, 1.39137306475090766519947664415, 3.86970191054242787116171827815, 4.34556229325705527960020470541, 5.28836651201910337578424608083, 6.32287864510958767972130452896, 7.00437659832491539446571500425, 8.247058292268234893871562936717, 8.673566143717895144725710520862, 9.833313917052550004323253796695

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