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Properties

Label 2-945-105.104-c1-0-28
Degree $2$
Conductor $945$
Sign $0.899 + 0.437i$
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  − 1.92·2-s + 1.70·4-s + (−1.22 + 1.86i)5-s + (2.62 − 0.341i)7-s + 0.572·8-s + (2.36 − 3.59i)10-s − 3.13i·11-s + 3.49·13-s + (−5.04 + 0.657i)14-s − 4.50·16-s + 0.661i·17-s − 4.05i·19-s + (−2.09 + 3.17i)20-s + 6.04i·22-s − 2.99·23-s + ⋯
L(s)  = 1  − 1.36·2-s + 0.851·4-s + (−0.550 + 0.835i)5-s + (0.991 − 0.129i)7-s + 0.202·8-s + (0.748 − 1.13i)10-s − 0.946i·11-s + 0.969·13-s + (−1.34 + 0.175i)14-s − 1.12·16-s + 0.160i·17-s − 0.930i·19-s + (−0.468 + 0.710i)20-s + 1.28i·22-s − 0.624·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.899 + 0.437i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (944, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.899 + 0.437i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.724742 - 0.166983i\)
\(L(\frac12)\) \(\approx\) \(0.724742 - 0.166983i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (1.22 - 1.86i)T \)
7 \( 1 + (-2.62 + 0.341i)T \)
good2 \( 1 + 1.92T + 2T^{2} \)
11 \( 1 + 3.13iT - 11T^{2} \)
13 \( 1 - 3.49T + 13T^{2} \)
17 \( 1 - 0.661iT - 17T^{2} \)
19 \( 1 + 4.05iT - 19T^{2} \)
23 \( 1 + 2.99T + 23T^{2} \)
29 \( 1 + 0.604iT - 29T^{2} \)
31 \( 1 + 1.99iT - 31T^{2} \)
37 \( 1 - 8.98iT - 37T^{2} \)
41 \( 1 - 2.09T + 41T^{2} \)
43 \( 1 + 8.55iT - 43T^{2} \)
47 \( 1 + 6.12iT - 47T^{2} \)
53 \( 1 - 5.42T + 53T^{2} \)
59 \( 1 - 5.81T + 59T^{2} \)
61 \( 1 + 4.13iT - 61T^{2} \)
67 \( 1 - 10.7iT - 67T^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 + 2.64T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 14.4iT - 83T^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 - 7.70T + 97T^{2} \)
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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

−10.17884162729772408442506160814, −8.831013138900689139307174297930, −8.461285925042384927750732111737, −7.69847609705013191180415321365, −6.93989878462190521261548538750, −5.95159043629817327669097002445, −4.57456378771572949917336113624, −3.50265471735045690593204275383, −2.11258964577219321419500532029, −0.70542297050790980852837341904, 1.08892220685629787982202763958, 1.98805342327314994089233064465, 3.97206393332251493265287114571, 4.72778063758435020669918557670, 5.84652894700030541969106158379, 7.24162373655086170075754250362, 7.86271501780946463665949352163, 8.458012603215479950630487549087, 9.123866406582131530483269600012, 9.928660741743494387514122417164

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