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Properties

Label 2-945-105.44-c0-0-1
Degree $2$
Conductor $945$
Sign $0.750 + 0.660i$
Analytic cond. $0.471616$
Root an. cond. $0.686743$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + i·7-s + (1.36 + 0.366i)10-s + i·13-s + (1.22 + 0.707i)14-s + (0.499 − 0.866i)16-s + (−0.707 − 1.22i)17-s + (0.707 − 0.707i)20-s + (−0.866 + 0.499i)25-s + (1.22 + 0.707i)26-s + (0.866 − 0.499i)28-s − 1.41i·29-s + (0.5 + 0.866i)31-s + (−0.707 − 1.22i)32-s + ⋯
L(s)  = 1  + (0.707 − 1.22i)2-s + (−0.499 − 0.866i)4-s + (0.258 + 0.965i)5-s + i·7-s + (1.36 + 0.366i)10-s + i·13-s + (1.22 + 0.707i)14-s + (0.499 − 0.866i)16-s + (−0.707 − 1.22i)17-s + (0.707 − 0.707i)20-s + (−0.866 + 0.499i)25-s + (1.22 + 0.707i)26-s + (0.866 − 0.499i)28-s − 1.41i·29-s + (0.5 + 0.866i)31-s + (−0.707 − 1.22i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 + 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.750 + 0.660i$
Analytic conductor: \(0.471616\)
Root analytic conductor: \(0.686743\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :0),\ 0.750 + 0.660i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.513819929\)
\(L(\frac12)\) \(\approx\) \(1.513819929\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
7 \( 1 - iT \)
good2 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 - iT - T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 + iT - T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 1.41iT - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + iT - T^{2} \)
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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

−10.41546898966904685048068168213, −9.534426106579060265394268587021, −8.854605760279948345413818016704, −7.46778418334834172865687145334, −6.64778577870711075898262702358, −5.61026411588875383121538949324, −4.66394051551017459447446902746, −3.61570251283594349792373301425, −2.58046918193338843595381900145, −2.03448533137599805147671048885, 1.46615935490813204297894292390, 3.50491652441461577475754100926, 4.49705585174785034525564854476, 5.08768417898253964749750475239, 6.10835644449775289388887898025, 6.71951976029342991607436181219, 7.955306743001593515168223761930, 8.146022706956392511500663995163, 9.364474323818639436425820748670, 10.36985745016371700649366006337

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