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Properties

Label 2-2300-115.22-c1-0-31
Degree $2$
Conductor $2300$
Sign $-0.342 + 0.939i$
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  + (1.58 − 1.58i)3-s + (0.216 − 0.216i)7-s − 2.03i·9-s − 3.54i·11-s + (−2.55 + 2.55i)13-s + (4.67 − 4.67i)17-s + 2.45·19-s − 0.687i·21-s + (−3.33 − 3.44i)23-s + (1.52 + 1.52i)27-s − 0.693i·29-s − 1.83·31-s + (−5.62 − 5.62i)33-s + (7.00 − 7.00i)37-s + 8.10i·39-s + ⋯
L(s)  = 1  + (0.916 − 0.916i)3-s + (0.0818 − 0.0818i)7-s − 0.679i·9-s − 1.06i·11-s + (−0.708 + 0.708i)13-s + (1.13 − 1.13i)17-s + 0.563·19-s − 0.150i·21-s + (−0.694 − 0.719i)23-s + (0.294 + 0.294i)27-s − 0.128i·29-s − 0.330·31-s + (−0.978 − 0.978i)33-s + (1.15 − 1.15i)37-s + 1.29i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $-0.342 + 0.939i$
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: \((2,\ 2300,\ (\ :1/2),\ -0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.278796315\)
\(L(\frac12)\) \(\approx\) \(2.278796315\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.33 + 3.44i)T \)
good3 \( 1 + (-1.58 + 1.58i)T - 3iT^{2} \)
7 \( 1 + (-0.216 + 0.216i)T - 7iT^{2} \)
11 \( 1 + 3.54iT - 11T^{2} \)
13 \( 1 + (2.55 - 2.55i)T - 13iT^{2} \)
17 \( 1 + (-4.67 + 4.67i)T - 17iT^{2} \)
19 \( 1 - 2.45T + 19T^{2} \)
29 \( 1 + 0.693iT - 29T^{2} \)
31 \( 1 + 1.83T + 31T^{2} \)
37 \( 1 + (-7.00 + 7.00i)T - 37iT^{2} \)
41 \( 1 + 2.34T + 41T^{2} \)
43 \( 1 + (4.16 + 4.16i)T + 43iT^{2} \)
47 \( 1 + (3.83 + 3.83i)T + 47iT^{2} \)
53 \( 1 + (0.724 + 0.724i)T + 53iT^{2} \)
59 \( 1 + 13.3iT - 59T^{2} \)
61 \( 1 - 7.37iT - 61T^{2} \)
67 \( 1 + (-5.28 + 5.28i)T - 67iT^{2} \)
71 \( 1 + 2.63T + 71T^{2} \)
73 \( 1 + (-4.02 + 4.02i)T - 73iT^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + (1.40 + 1.40i)T + 83iT^{2} \)
89 \( 1 + 2.83T + 89T^{2} \)
97 \( 1 + (-3.72 + 3.72i)T - 97iT^{2} \)
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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

−8.659627800799326770919046846994, −7.895416851039611626174041474782, −7.42571080136048538372286216404, −6.65649924148456508732124779672, −5.70200234291917261555413896410, −4.79934703573140794724098859045, −3.57987631051323225025325938586, −2.81157372213967517507003722187, −1.93446421257461940838020693635, −0.69368366221174641608777240416, 1.54111605681945863868466247574, 2.74704434529105221430798970494, 3.48819068064906533774245336566, 4.32125431073163270716578592283, 5.12148848298084175727957688766, 5.98109506452926303789044903317, 7.15257543433397077295620496424, 7.936156717460408103133977153664, 8.385509167911848506642029500825, 9.467317330560087316041517084197

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