Properties

Label 2-300-60.59-c1-0-14
Degree $2$
Conductor $300$
Sign $0.369 - 0.929i$
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  + (1.38 + 0.273i)2-s + (−0.758 + 1.55i)3-s + (1.85 + 0.758i)4-s + (−1.47 + 1.95i)6-s + 3.56·7-s + (2.36 + 1.55i)8-s + (−1.85 − 2.36i)9-s − 4.20·11-s + (−2.58 + 2.30i)12-s + 2.70i·13-s + (4.94 + 0.973i)14-s + (2.85 + 2.80i)16-s − 0.828·17-s + (−1.92 − 3.78i)18-s − 5.07i·19-s + ⋯
L(s)  = 1  + (0.981 + 0.193i)2-s + (−0.437 + 0.899i)3-s + (0.925 + 0.379i)4-s + (−0.603 + 0.797i)6-s + 1.34·7-s + (0.834 + 0.550i)8-s + (−0.616 − 0.787i)9-s − 1.26·11-s + (−0.745 + 0.666i)12-s + 0.749i·13-s + (1.32 + 0.260i)14-s + (0.712 + 0.701i)16-s − 0.200·17-s + (−0.453 − 0.891i)18-s − 1.16i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.369 - 0.929i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.369 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76690 + 1.19939i\)
\(L(\frac12)\) \(\approx\) \(1.76690 + 1.19939i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−11.72908610313422564896024536689, −11.11393928178143066169607920302, −10.52206695555788771647671451950, −9.060047218875189890482048309876, −7.962258557206739856742590641867, −6.88045712106366510532647836014, −5.45717496684029536664096964051, −4.94953225706246772417277895058, −3.95357016974184294017591657227, −2.36847889215001632135701481104, 1.55348969953339636927546846569, 2.83637714172766041345201897268, 4.70682675369356361473083816918, 5.42025789886135677619598138571, 6.41087661817973823919429553520, 7.85864231390870039467403343512, 8.006916917198646344640395342498, 10.24022997608290978830643659226, 10.92582012820310861262663055158, 11.71254422044018412096208783842

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