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Properties

Label 2-945-21.20-c1-0-24
Degree $2$
Conductor $945$
Sign $0.782 - 0.623i$
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  + 2.67i·2-s − 5.13·4-s − 5-s + (1.64 + 2.06i)7-s − 8.37i·8-s − 2.67i·10-s − 4.55i·11-s − 4.75i·13-s + (−5.52 + 4.40i)14-s + 12.0·16-s − 2.70·17-s − 6.55i·19-s + 5.13·20-s + 12.1·22-s + 6.74i·23-s + ⋯
L(s)  = 1  + 1.88i·2-s − 2.56·4-s − 0.447·5-s + (0.623 + 0.782i)7-s − 2.95i·8-s − 0.844i·10-s − 1.37i·11-s − 1.31i·13-s + (−1.47 + 1.17i)14-s + 3.02·16-s − 0.656·17-s − 1.50i·19-s + 1.14·20-s + 2.59·22-s + 1.40i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.782 - 0.623i$
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: \((2,\ 945,\ (\ :1/2),\ 0.782 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.851753 + 0.297782i\)
\(L(\frac12)\) \(\approx\) \(0.851753 + 0.297782i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 - 2.06i)T \)
good2 \( 1 - 2.67iT - 2T^{2} \)
11 \( 1 + 4.55iT - 11T^{2} \)
13 \( 1 + 4.75iT - 13T^{2} \)
17 \( 1 + 2.70T + 17T^{2} \)
19 \( 1 + 6.55iT - 19T^{2} \)
23 \( 1 - 6.74iT - 23T^{2} \)
29 \( 1 + 3.19iT - 29T^{2} \)
31 \( 1 + 7.05iT - 31T^{2} \)
37 \( 1 - 8.66T + 37T^{2} \)
41 \( 1 + 4.62T + 41T^{2} \)
43 \( 1 - 3.29T + 43T^{2} \)
47 \( 1 + 9.69T + 47T^{2} \)
53 \( 1 + 3.37iT - 53T^{2} \)
59 \( 1 + 2.55T + 59T^{2} \)
61 \( 1 + 4.84iT - 61T^{2} \)
67 \( 1 + 2.18T + 67T^{2} \)
71 \( 1 - 1.61iT - 71T^{2} \)
73 \( 1 - 9.66iT - 73T^{2} \)
79 \( 1 + 1.73T + 79T^{2} \)
83 \( 1 - 5.20T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 + 0.365iT - 97T^{2} \)
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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

−9.607701372582636186934670014351, −8.950123520610231502008530860325, −8.067748790379855932582714540027, −7.87281317008014252626226802538, −6.68875789471167472890483048137, −5.80359541478753281905886950567, −5.28517105215080664884020512189, −4.30468021624428633226204885297, −3.04118466177194569471920413613, −0.46760739941761218803402683003, 1.38492074986746613696887309274, 2.19539538299858322217643340220, 3.59936960544394684994821584347, 4.47912161130804001097924228608, 4.75197999007440710112383811109, 6.62304599541458221631338596433, 7.72010864931441189147622423398, 8.607263757105227482081749217920, 9.410206902961893233720342767837, 10.28249846886171111202157784224

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