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Properties

Label 2-2303-2303.1268-c0-0-4
Degree $2$
Conductor $2303$
Sign $-0.949 + 0.315i$
Analytic cond. $1.14934$
Root an. cond. $1.07207$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.24 − 1.56i)2-s + (0.400 + 0.193i)3-s + (−0.667 − 2.92i)4-s + (0.801 − 0.386i)6-s + (0.623 + 0.781i)7-s + (−3.60 − 1.73i)8-s + (−0.499 − 0.626i)9-s + (0.297 − 1.30i)12-s + 1.99·14-s + (−4.50 + 2.16i)16-s + (0.400 − 1.75i)17-s − 1.60·18-s + (0.0990 + 0.433i)21-s + (−1.10 − 1.39i)24-s + (0.623 + 0.781i)25-s + ⋯
L(s)  = 1  + (1.24 − 1.56i)2-s + (0.400 + 0.193i)3-s + (−0.667 − 2.92i)4-s + (0.801 − 0.386i)6-s + (0.623 + 0.781i)7-s + (−3.60 − 1.73i)8-s + (−0.499 − 0.626i)9-s + (0.297 − 1.30i)12-s + 1.99·14-s + (−4.50 + 2.16i)16-s + (0.400 − 1.75i)17-s − 1.60·18-s + (0.0990 + 0.433i)21-s + (−1.10 − 1.39i)24-s + (0.623 + 0.781i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2303 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 + 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2303\)    =    \(7^{2} \cdot 47\)
Sign: $-0.949 + 0.315i$
Analytic conductor: \(1.14934\)
Root analytic conductor: \(1.07207\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2303} (1268, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2303,\ (\ :0),\ -0.949 + 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.346453653\)
\(L(\frac12)\) \(\approx\) \(2.346453653\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.623 - 0.781i)T \)
47 \( 1 + (-0.623 + 0.781i)T \)
good2 \( 1 + (-1.24 + 1.56i)T + (-0.222 - 0.974i)T^{2} \)
3 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
5 \( 1 + (-0.623 - 0.781i)T^{2} \)
11 \( 1 + (0.222 + 0.974i)T^{2} \)
13 \( 1 + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.900 - 0.433i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
59 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
61 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.0990 - 0.433i)T + (-0.900 + 0.433i)T^{2} \)
73 \( 1 + (0.222 - 0.974i)T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
83 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (-1.24 - 1.56i)T + (-0.222 + 0.974i)T^{2} \)
97 \( 1 - 2T + T^{2} \)
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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

−9.118521887354906957683511740295, −8.577707496389920245440306257211, −7.07018716313031097170935929876, −6.04959655917946239336586221334, −5.28589562016610378830529681673, −4.80088848626660053491050242149, −3.72864768279253272619174654480, −2.94827031642741582147148331308, −2.38069601909748078817320437989, −1.09081664922212956172201854934, 2.21387375691438495893303040211, 3.38426777983114657792666153306, 4.09626800810204755218676070938, 4.85790274523452821788628963552, 5.66723388591512122646155215214, 6.34494844176976943785411729988, 7.29022997831798359338940749990, 7.75740237338050149559850332704, 8.432892007302744779625874109071, 8.861407536161869964396853382805

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