Properties

Label 2-105-1.1-c1-0-1
Degree $2$
Conductor $105$
Sign $1$
Analytic cond. $0.838429$
Root an. cond. $0.915657$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s − 4-s + 5-s + 6-s + 7-s − 3·8-s + 9-s + 10-s − 12-s − 6·13-s + 14-s + 15-s − 16-s + 2·17-s + 18-s − 8·19-s − 20-s + 21-s + 8·23-s − 3·24-s + 25-s − 6·26-s + 27-s − 28-s − 2·29-s + 30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 1.66·13-s + 0.267·14-s + 0.258·15-s − 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.218·21-s + 1.66·23-s − 0.612·24-s + 1/5·25-s − 1.17·26-s + 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.182·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(105\)    =    \(3 \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(0.838429\)
Root analytic conductor: \(0.915657\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{105} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 105,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.465863363\)
\(L(\frac12)\) \(\approx\) \(1.465863363\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 - T \)
7 \( 1 - T \)
good2 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.78714664919275649975914854040, −12.89916852888262332222898282203, −12.11417775373724942408035215331, −10.47975989489334108877507822892, −9.386500711260631787757553076630, −8.447489066361171687346291758504, −6.97944382340517249810347443111, −5.37573342552736399524296425305, −4.33309058948755389799390879771, −2.63294756479994443235170715895, 2.63294756479994443235170715895, 4.33309058948755389799390879771, 5.37573342552736399524296425305, 6.97944382340517249810347443111, 8.447489066361171687346291758504, 9.386500711260631787757553076630, 10.47975989489334108877507822892, 12.11417775373724942408035215331, 12.89916852888262332222898282203, 13.78714664919275649975914854040

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