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Properties

Label 2-1932-7.4-c1-0-15
Degree $2$
Conductor $1932$
Sign $0.993 + 0.112i$
Analytic cond. $15.4270$
Root an. cond. $3.92773$
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  + (−0.5 + 0.866i)3-s + (−0.118 − 0.205i)5-s + (2.35 − 1.20i)7-s + (−0.499 − 0.866i)9-s − 2.19·13-s + 0.237·15-s + (−0.480 + 0.831i)17-s + (0.933 + 1.61i)19-s + (−0.129 + 2.64i)21-s + (0.5 + 0.866i)23-s + (2.47 − 4.28i)25-s + 0.999·27-s + 5.27·29-s + (3.83 − 6.63i)31-s + (−0.527 − 0.339i)35-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.0529 − 0.0917i)5-s + (0.889 − 0.456i)7-s + (−0.166 − 0.288i)9-s − 0.609·13-s + 0.0611·15-s + (−0.116 + 0.201i)17-s + (0.214 + 0.370i)19-s + (−0.0283 + 0.576i)21-s + (0.104 + 0.180i)23-s + (0.494 − 0.856i)25-s + 0.192·27-s + 0.979·29-s + (0.688 − 1.19i)31-s + (−0.0890 − 0.0574i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1932 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1932\)    =    \(2^{2} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.993 + 0.112i$
Analytic conductor: \(15.4270\)
Root analytic conductor: \(3.92773\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1932} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1932,\ (\ :1/2),\ 0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.667819100\)
\(L(\frac12)\) \(\approx\) \(1.667819100\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-2.35 + 1.20i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.118 + 0.205i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2.19T + 13T^{2} \)
17 \( 1 + (0.480 - 0.831i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.933 - 1.61i)T + (-9.5 + 16.4i)T^{2} \)
29 \( 1 - 5.27T + 29T^{2} \)
31 \( 1 + (-3.83 + 6.63i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.76 - 6.51i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.79T + 41T^{2} \)
43 \( 1 + 1.86T + 43T^{2} \)
47 \( 1 + (5.32 + 9.23i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.24 - 7.35i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.63 - 4.56i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.84 + 8.39i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.05 + 5.29i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + (-5.67 + 9.83i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.53 - 4.39i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.78T + 83T^{2} \)
89 \( 1 + (-3.76 - 6.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7.63T + 97T^{2} \)
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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

−9.289512869177535220387484410162, −8.239351039123155178943234258649, −7.83654123314531332529169729059, −6.74859096928944498451007289630, −5.97744048092872228078681553654, −4.81634563948388612014938273790, −4.56124357709698124361509238854, −3.40963154458557692247892147686, −2.20259904718492823828162569404, −0.808692253285143044057728979877, 1.02109709474044461788384585971, 2.20405592191048089463608970185, 3.11326972342873201200815631976, 4.60776927291662296942115692853, 5.07811553977786620689207279304, 6.05451755701035856611406814648, 6.91873093281045059496286576591, 7.61448221772044905374027486458, 8.360185640957154445147993327494, 9.100788861176718364373401713377

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