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Properties

Label 2-372-3.2-c2-0-2
Degree $2$
Conductor $372$
Sign $-0.114 - 0.993i$
Analytic cond. $10.1362$
Root an. cond. $3.18375$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.344 − 2.98i)3-s + 7.78i·5-s + 6.16·7-s + (−8.76 + 2.05i)9-s + 0.942i·11-s − 21.7·13-s + (23.2 − 2.68i)15-s + 30.8i·17-s − 22.6·19-s + (−2.12 − 18.3i)21-s − 0.406i·23-s − 35.6·25-s + (9.14 + 25.4i)27-s − 6.48i·29-s + 5.56·31-s + ⋯
L(s)  = 1  + (−0.114 − 0.993i)3-s + 1.55i·5-s + 0.880·7-s + (−0.973 + 0.228i)9-s + 0.0856i·11-s − 1.67·13-s + (1.54 − 0.179i)15-s + 1.81i·17-s − 1.19·19-s + (−0.101 − 0.874i)21-s − 0.0176i·23-s − 1.42·25-s + (0.338 + 0.940i)27-s − 0.223i·29-s + 0.179·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 372 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.114 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(372\)    =    \(2^{2} \cdot 3 \cdot 31\)
Sign: $-0.114 - 0.993i$
Analytic conductor: \(10.1362\)
Root analytic conductor: \(3.18375\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{372} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 372,\ (\ :1),\ -0.114 - 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.693110 + 0.777942i\)
\(L(\frac12)\) \(\approx\) \(0.693110 + 0.777942i\)
\(L(2)\) 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 \( 1 + (0.344 + 2.98i)T \)
31 \( 1 - 5.56T \)
good5 \( 1 - 7.78iT - 25T^{2} \)
7 \( 1 - 6.16T + 49T^{2} \)
11 \( 1 - 0.942iT - 121T^{2} \)
13 \( 1 + 21.7T + 169T^{2} \)
17 \( 1 - 30.8iT - 289T^{2} \)
19 \( 1 + 22.6T + 361T^{2} \)
23 \( 1 + 0.406iT - 529T^{2} \)
29 \( 1 + 6.48iT - 841T^{2} \)
37 \( 1 - 45.7T + 1.36e3T^{2} \)
41 \( 1 - 65.2iT - 1.68e3T^{2} \)
43 \( 1 - 43.6T + 1.84e3T^{2} \)
47 \( 1 - 77.0iT - 2.20e3T^{2} \)
53 \( 1 + 40.0iT - 2.80e3T^{2} \)
59 \( 1 + 52.7iT - 3.48e3T^{2} \)
61 \( 1 + 40.6T + 3.72e3T^{2} \)
67 \( 1 + 0.612T + 4.48e3T^{2} \)
71 \( 1 - 14.5iT - 5.04e3T^{2} \)
73 \( 1 - 71.8T + 5.32e3T^{2} \)
79 \( 1 + 78.3T + 6.24e3T^{2} \)
83 \( 1 + 67.0iT - 6.88e3T^{2} \)
89 \( 1 - 46.3iT - 7.92e3T^{2} \)
97 \( 1 + 60.0T + 9.40e3T^{2} \)
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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

−11.25074308971294544935106713894, −10.81074326405328799005748573386, −9.767341838424176999661198632387, −8.184603157322210438140083547557, −7.67910075200190712277126975595, −6.67273000821377957319410706151, −5.98950628694424536126662180257, −4.46480072211681344028590720638, −2.79999580834102138753760808172, −1.90164162911177640544765630451, 0.44898303178698182268388627023, 2.42356755048062815739961478093, 4.32780735389841422376396026146, 4.85474951625105249815602797235, 5.55535077611578905388917768462, 7.33643868830818864228697175541, 8.405717293040396604597467945479, 9.149755605334420611576524648905, 9.814811778472625080114500917169, 10.94415390937369223650183204119

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