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Properties

Label 2-2300-115.22-c1-0-19
Degree $2$
Conductor $2300$
Sign $0.973 - 0.229i$
Analytic cond. $18.3655$
Root an. cond. $4.28550$
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  + (0.326 − 0.326i)7-s + 3i·9-s + (5.80 − 5.80i)17-s + (3.39 + 3.39i)23-s − 9.78i·29-s − 7.78·31-s + (−5.15 + 5.15i)37-s + 5.78·41-s + (6.78 + 6.78i)43-s + 6.78i·49-s + (6.45 + 6.45i)53-s + 3.78i·59-s + (0.978 + 0.978i)63-s + (11.2 − 11.2i)67-s + 15.7·71-s + ⋯
L(s)  = 1  + (0.123 − 0.123i)7-s + i·9-s + (1.40 − 1.40i)17-s + (0.707 + 0.707i)23-s − 1.81i·29-s − 1.39·31-s + (−0.846 + 0.846i)37-s + 0.903·41-s + (1.03 + 1.03i)43-s + 0.969i·49-s + (0.886 + 0.886i)53-s + 0.493i·59-s + (0.123 + 0.123i)63-s + (1.37 − 1.37i)67-s + 1.87·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(18.3655\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2300} (1057, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :1/2),\ 0.973 - 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.870437352\)
\(L(\frac12)\) \(\approx\) \(1.870437352\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 + (-3.39 - 3.39i)T \)
good3 \( 1 - 3iT^{2} \)
7 \( 1 + (-0.326 + 0.326i)T - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-5.80 + 5.80i)T - 17iT^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 + 9.78iT - 29T^{2} \)
31 \( 1 + 7.78T + 31T^{2} \)
37 \( 1 + (5.15 - 5.15i)T - 37iT^{2} \)
41 \( 1 - 5.78T + 41T^{2} \)
43 \( 1 + (-6.78 - 6.78i)T + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-6.45 - 6.45i)T + 53iT^{2} \)
59 \( 1 - 3.78iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + (-11.2 + 11.2i)T - 67iT^{2} \)
71 \( 1 - 15.7T + 71T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + (-11.9 - 11.9i)T + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + (-13.5 + 13.5i)T - 97iT^{2} \)
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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

−9.269969192738882606449397166453, −7.88313233019266132852983029606, −7.76789815455283110988920497966, −6.85315521508900145827030189671, −5.68109720505758689127978578563, −5.18469529238934998963374480486, −4.25861446366551441865792437157, −3.17158860416504531602786579478, −2.26737674917802632965530174721, −0.959570048887910862608781447111, 0.870080839556439329403905544212, 2.05165158863438271313891132369, 3.45652769601775072337787395763, 3.81863702715107650226098751891, 5.21492992858604399453594018939, 5.73752072178130729369635602530, 6.74039444571544542724631698719, 7.32464975797547966432455611330, 8.381801062594564243121034828640, 8.902403290592334652646003434559

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