Properties

Label 2-300-12.11-c1-0-26
Degree $2$
Conductor $300$
Sign $-0.666 + 0.745i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
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  + (−0.273 − 1.38i)2-s + (1.55 − 0.758i)3-s + (−1.85 + 0.758i)4-s + (−1.47 − 1.95i)6-s − 3.56i·7-s + (1.55 + 2.36i)8-s + (1.85 − 2.36i)9-s − 4.20·11-s + (−2.30 + 2.58i)12-s + 2.70·13-s + (−4.94 + 0.973i)14-s + (2.85 − 2.80i)16-s + 0.828i·17-s + (−3.78 − 1.92i)18-s − 5.07i·19-s + ⋯
L(s)  = 1  + (−0.193 − 0.981i)2-s + (0.899 − 0.437i)3-s + (−0.925 + 0.379i)4-s + (−0.603 − 0.797i)6-s − 1.34i·7-s + (0.550 + 0.834i)8-s + (0.616 − 0.787i)9-s − 1.26·11-s + (−0.666 + 0.745i)12-s + 0.749·13-s + (−1.32 + 0.260i)14-s + (0.712 − 0.701i)16-s + 0.200i·17-s + (−0.891 − 0.453i)18-s − 1.16i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.666 + 0.745i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.666 + 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557987 - 1.24661i\)
\(L(\frac12)\) \(\approx\) \(0.557987 - 1.24661i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.273 + 1.38i)T \)
3 \( 1 + (-1.55 + 0.758i)T \)
5 \( 1 \)
good7 \( 1 + 3.56iT - 7T^{2} \)
11 \( 1 + 4.20T + 11T^{2} \)
13 \( 1 - 2.70T + 13T^{2} \)
17 \( 1 - 0.828iT - 17T^{2} \)
19 \( 1 + 5.07iT - 19T^{2} \)
23 \( 1 - 1.09T + 23T^{2} \)
29 \( 1 - 5.55iT - 29T^{2} \)
31 \( 1 - 6.59iT - 31T^{2} \)
37 \( 1 - 5.40T + 37T^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + 0.531iT - 43T^{2} \)
47 \( 1 - 6.22T + 47T^{2} \)
53 \( 1 + 5.55iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 0.701T + 61T^{2} \)
67 \( 1 - 2.04iT - 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 - 7.70T + 73T^{2} \)
79 \( 1 - 7.12iT - 79T^{2} \)
83 \( 1 - 3.11T + 83T^{2} \)
89 \( 1 + 4.72iT - 89T^{2} \)
97 \( 1 + 8.10T + 97T^{2} \)
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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

−11.14286366929421312642585109259, −10.55327570140271168810577759571, −9.618394477459269690648487101708, −8.573377753536203383256367974673, −7.80463919377036841263323981193, −6.85610317253940320229624868625, −4.89752674401541881697235875089, −3.72977384243359151588540675288, −2.71306679449250231284640212447, −1.08287917129573480032568601736, 2.39910027569157357029289583304, 3.93552589258612856189814586927, 5.27250910433095717077873714338, 6.03047721130242110340487955725, 7.64064624213978391399390823198, 8.223272668661541289103430197430, 9.063993060497432910770503270068, 9.846693819763601134270087611419, 10.83149545921053663709718687619, 12.36204068638758635166015882750

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