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Properties

Label 2-936-8.5-c1-0-23
Degree $2$
Conductor $936$
Sign $-0.707 - 0.707i$
Analytic cond. $7.47399$
Root an. cond. $2.73386$
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 + i)2-s + 2i·4-s + 3i·5-s + 3·7-s + (−2 + 2i)8-s + (−3 + 3i)10-s − i·13-s + (3 + 3i)14-s − 4·16-s + 7·17-s + 4i·19-s − 6·20-s − 4·23-s − 4·25-s + (1 − i)26-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + i·4-s + 1.34i·5-s + 1.13·7-s + (−0.707 + 0.707i)8-s + (−0.948 + 0.948i)10-s − 0.277i·13-s + (0.801 + 0.801i)14-s − 16-s + 1.69·17-s + 0.917i·19-s − 1.34·20-s − 0.834·23-s − 0.800·25-s + (0.196 − 0.196i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(936\)    =    \(2^{3} \cdot 3^{2} \cdot 13\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(7.47399\)
Root analytic conductor: \(2.73386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{936} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 936,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953463 + 2.30186i\)
\(L(\frac12)\) \(\approx\) \(0.953463 + 2.30186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
17 \( 1 - 7T + 17T^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + 7iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 - 7T + 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 + 14iT - 59T^{2} \)
61 \( 1 - 10iT - 61T^{2} \)
67 \( 1 + 2iT - 67T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−10.54879325857465538070230369061, −9.568344033548428929447615449757, −8.189352302905368400678882268362, −7.74795126446030632420585194532, −7.04750305861168250111128132207, −5.90549497669378764643639949139, −5.41375683031812336635310272242, −4.06336662775372832913137138087, −3.30337636812120140812732157124, −2.07875543361596919789728301552, 1.01251428109686507359472261915, 1.91542847886084698137045959320, 3.45500436521338596356623048772, 4.54168481132733101164581341578, 5.12351612177070882626560888552, 5.80182269904285993340750573937, 7.23009806150085706545676071888, 8.256892312600616901149994718986, 9.002976634679524268556987485372, 9.810929708715935350486376354643

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