Properties

Label 2-350-1.1-c1-0-8
Degree $2$
Conductor $350$
Sign $1$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2.44·3-s + 4-s + 2.44·6-s + 7-s + 8-s + 2.99·9-s − 4.89·11-s + 2.44·12-s − 4.44·13-s + 14-s + 16-s − 2·17-s + 2.99·18-s + 1.55·19-s + 2.44·21-s − 4.89·22-s − 2.89·23-s + 2.44·24-s − 4.44·26-s + 28-s + 6.89·29-s + 8.89·31-s + 32-s − 11.9·33-s − 2·34-s + 2.99·36-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.41·3-s + 0.5·4-s + 0.999·6-s + 0.377·7-s + 0.353·8-s + 0.999·9-s − 1.47·11-s + 0.707·12-s − 1.23·13-s + 0.267·14-s + 0.250·16-s − 0.485·17-s + 0.707·18-s + 0.355·19-s + 0.534·21-s − 1.04·22-s − 0.604·23-s + 0.499·24-s − 0.872·26-s + 0.188·28-s + 1.28·29-s + 1.59·31-s + 0.176·32-s − 2.08·33-s − 0.342·34-s + 0.499·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.830982530\)
\(L(\frac12)\) \(\approx\) \(2.830982530\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
good3 \( 1 - 2.44T + 3T^{2} \)
11 \( 1 + 4.89T + 11T^{2} \)
13 \( 1 + 4.44T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 1.55T + 19T^{2} \)
23 \( 1 + 2.89T + 23T^{2} \)
29 \( 1 - 6.89T + 29T^{2} \)
31 \( 1 - 8.89T + 31T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + 1.10T + 41T^{2} \)
43 \( 1 - 0.898T + 43T^{2} \)
47 \( 1 + 8.89T + 47T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + 1.55T + 59T^{2} \)
61 \( 1 - 3.55T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 1.10T + 71T^{2} \)
73 \( 1 + 2.89T + 73T^{2} \)
79 \( 1 - 6.89T + 79T^{2} \)
83 \( 1 - 2.44T + 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.70456166423393402195005342402, −10.40695359032285207903865428901, −9.738844466819335119483763061725, −8.375924916232554564968780107376, −7.88805460590994197147068727888, −6.86512930623403544071985121749, −5.30743194698470000684988949513, −4.39664725359770703549679315738, −2.95437356862933556349589288599, −2.28923518601867728793453841962, 2.28923518601867728793453841962, 2.95437356862933556349589288599, 4.39664725359770703549679315738, 5.30743194698470000684988949513, 6.86512930623403544071985121749, 7.88805460590994197147068727888, 8.375924916232554564968780107376, 9.738844466819335119483763061725, 10.40695359032285207903865428901, 11.70456166423393402195005342402

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