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Properties

Label 2-945-21.20-c1-0-2
Degree $2$
Conductor $945$
Sign $-0.00246 + 0.999i$
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  + 2.76i·2-s − 5.61·4-s + 5-s + (−0.00652 + 2.64i)7-s − 9.98i·8-s + 2.76i·10-s + 4.51i·11-s + 3.02i·13-s + (−7.30 − 0.0180i)14-s + 16.3·16-s − 5.34·17-s + 2.84i·19-s − 5.61·20-s − 12.4·22-s − 2.20i·23-s + ⋯
L(s)  = 1  + 1.95i·2-s − 2.80·4-s + 0.447·5-s + (−0.00246 + 0.999i)7-s − 3.53i·8-s + 0.872i·10-s + 1.36i·11-s + 0.838i·13-s + (−1.95 − 0.00481i)14-s + 4.08·16-s − 1.29·17-s + 0.652i·19-s − 1.25·20-s − 2.65·22-s − 0.458i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00246 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00246 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.00246 + 0.999i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (566, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.00246 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.606732 - 0.608232i\)
\(L(\frac12)\) \(\approx\) \(0.606732 - 0.608232i\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + (0.00652 - 2.64i)T \)
good2 \( 1 - 2.76iT - 2T^{2} \)
11 \( 1 - 4.51iT - 11T^{2} \)
13 \( 1 - 3.02iT - 13T^{2} \)
17 \( 1 + 5.34T + 17T^{2} \)
19 \( 1 - 2.84iT - 19T^{2} \)
23 \( 1 + 2.20iT - 23T^{2} \)
29 \( 1 + 5.09iT - 29T^{2} \)
31 \( 1 + 7.68iT - 31T^{2} \)
37 \( 1 - 1.38T + 37T^{2} \)
41 \( 1 + 9.70T + 41T^{2} \)
43 \( 1 - 0.136T + 43T^{2} \)
47 \( 1 + 0.149T + 47T^{2} \)
53 \( 1 - 5.09iT - 53T^{2} \)
59 \( 1 - 1.39T + 59T^{2} \)
61 \( 1 - 7.68iT - 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 10.8iT - 71T^{2} \)
73 \( 1 - 7.91iT - 73T^{2} \)
79 \( 1 + 1.95T + 79T^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + 13.6T + 89T^{2} \)
97 \( 1 - 8.74iT - 97T^{2} \)
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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

−10.05941786181423420088782434804, −9.503851364285593205299436213830, −8.831137350982563552710192039289, −8.088127569397824490588083513550, −7.10021649430130150475254578753, −6.45153924767302022910829250193, −5.75459099390998641488632973838, −4.77392754958821126717685436164, −4.14134059734422279089266778711, −2.13500151046722348986344335734, 0.41481380802348640032160048569, 1.57090520681572278545319982149, 2.94343561669086461316915907939, 3.55991417951035971293256479193, 4.69048433469412271585897390331, 5.48152799984546961870432655735, 6.86684339658749464145786864442, 8.345064504960040956714867440674, 8.762828952488834442992560084804, 9.787931009652009422132029087032

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