Properties

Label 2-300-12.11-c1-0-22
Degree $2$
Conductor $300$
Sign $0.982 + 0.188i$
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  + (1.29 − 0.569i)2-s + (0.908 + 1.47i)3-s + (1.35 − 1.47i)4-s + (2.01 + 1.39i)6-s − 2.50i·7-s + (0.908 − 2.67i)8-s + (−1.35 + 2.67i)9-s + 3.36·11-s + (3.40 + 0.652i)12-s − 3.70·13-s + (−1.42 − 3.24i)14-s + (−0.350 − 3.98i)16-s + 7.63i·17-s + (−0.222 + 4.23i)18-s + 0.440i·19-s + ⋯
L(s)  = 1  + (0.915 − 0.402i)2-s + (0.524 + 0.851i)3-s + (0.675 − 0.737i)4-s + (0.822 + 0.568i)6-s − 0.948i·7-s + (0.321 − 0.947i)8-s + (−0.450 + 0.892i)9-s + 1.01·11-s + (0.982 + 0.188i)12-s − 1.02·13-s + (−0.382 − 0.868i)14-s + (−0.0876 − 0.996i)16-s + 1.85i·17-s + (−0.0523 + 0.998i)18-s + 0.100i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.982 + 0.188i$
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: \((2,\ 300,\ (\ :1/2),\ 0.982 + 0.188i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.46038 - 0.233976i\)
\(L(\frac12)\) \(\approx\) \(2.46038 - 0.233976i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.29 + 0.569i)T \)
3 \( 1 + (-0.908 - 1.47i)T \)
5 \( 1 \)
good7 \( 1 + 2.50iT - 7T^{2} \)
11 \( 1 - 3.36T + 11T^{2} \)
13 \( 1 + 3.70T + 13T^{2} \)
17 \( 1 - 7.63iT - 17T^{2} \)
19 \( 1 - 0.440iT - 19T^{2} \)
23 \( 1 + 5.17T + 23T^{2} \)
29 \( 1 - 2.27iT - 29T^{2} \)
31 \( 1 + 3.39iT - 31T^{2} \)
37 \( 1 + 7.40T + 37T^{2} \)
41 \( 1 + 3.07iT - 41T^{2} \)
43 \( 1 + 8.40iT - 43T^{2} \)
47 \( 1 - 3.63T + 47T^{2} \)
53 \( 1 + 2.27iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 5.70T + 61T^{2} \)
67 \( 1 - 5.45iT - 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 - 1.29T + 73T^{2} \)
79 \( 1 - 5.01iT - 79T^{2} \)
83 \( 1 - 1.81T + 83T^{2} \)
89 \( 1 - 5.35iT - 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−11.77127162290808817772336088701, −10.54637760624169797655578866450, −10.25572536099128367037625449028, −9.136932183509172917912092486390, −7.81966598818612303578450855724, −6.65865152888895422591626883063, −5.41589886492906235128910511328, −4.12265426821923548189534241450, −3.71669284931644949903297781325, −1.99826884252279656142511927702, 2.16702125005660435668153660174, 3.18540865369009122824476435331, 4.73677071918912998673428356577, 5.91659893010249290172922113870, 6.86293551607095883892232185877, 7.64946749837842434636664388534, 8.750781648075240756217504738259, 9.606549294343419105088409090327, 11.49419167244738335609795201736, 12.04353333792620796636048331901

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