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Properties

Label 2-945-945.839-c1-0-62
Degree $2$
Conductor $945$
Sign $0.954 - 0.296i$
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  + (−0.306 − 1.70i)3-s + (1.87 + 0.684i)4-s + (1.43 + 1.71i)5-s + (2.48 − 0.904i)7-s + (−2.81 + 1.04i)9-s + (−2.55 + 3.05i)11-s + (0.589 − 3.41i)12-s + (1.10 + 6.24i)13-s + (2.47 − 2.97i)15-s + (3.06 + 2.57i)16-s + (4.17 − 2.40i)17-s + (1.52 + 4.20i)20-s + (−2.30 − 3.96i)21-s + (−0.868 + 4.92i)25-s + (2.64 + 4.47i)27-s + 5.29·28-s + ⋯
L(s)  = 1  + (−0.177 − 0.984i)3-s + (0.939 + 0.342i)4-s + (0.642 + 0.766i)5-s + (0.939 − 0.342i)7-s + (−0.937 + 0.348i)9-s + (−0.771 + 0.919i)11-s + (0.170 − 0.985i)12-s + (0.305 + 1.73i)13-s + (0.640 − 0.768i)15-s + (0.766 + 0.642i)16-s + (1.01 − 0.584i)17-s + (0.342 + 0.939i)20-s + (−0.503 − 0.864i)21-s + (−0.173 + 0.984i)25-s + (0.509 + 0.860i)27-s + 0.999·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.954 - 0.296i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.954 - 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15170 + 0.326866i\)
\(L(\frac12)\) \(\approx\) \(2.15170 + 0.326866i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.306 + 1.70i)T \)
5 \( 1 + (-1.43 - 1.71i)T \)
7 \( 1 + (-2.48 + 0.904i)T \)
good2 \( 1 + (-1.87 - 0.684i)T^{2} \)
11 \( 1 + (2.55 - 3.05i)T + (-1.91 - 10.8i)T^{2} \)
13 \( 1 + (-1.10 - 6.24i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-4.17 + 2.40i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (10.5 + 1.86i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (4.67 + 12.8i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-6.07 + 3.50i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-7.26 + 12.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.00 + 17.0i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (-13.8 - 2.44i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.5 + 9.71i)T + (16.8 + 95.5i)T^{2} \)
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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

−10.30285442072640503963848144235, −9.271331187935784351860160917614, −8.003712872382604252867137943463, −7.34482774720885370040161196113, −6.94224928858041289801472011524, −5.99727984837538284862119686133, −5.05145265377353330306725505594, −3.49803979240086339564379788638, −2.09441054454937508727227152675, −1.82375399374400603077507802689, 1.09298685677200823795444311441, 2.55481751486213189471359821852, 3.58498390855819044876026582575, 5.21545024646986713576978585719, 5.47776359959746509018628371282, 6.07526945450359505206553880128, 7.899184844443548541157404001127, 8.188034545790743522525377874664, 9.327471919918029001623365519093, 10.15502881526181248453044283689

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