Properties

Label 2-35e2-1.1-c1-0-17
Degree $2$
Conductor $1225$
Sign $1$
Analytic cond. $9.78167$
Root an. cond. $3.12756$
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  − 1.91·2-s + 1.65·3-s + 1.65·4-s − 3.16·6-s + 0.656·8-s − 0.255·9-s + 4.48·11-s + 2.74·12-s + 6.48·13-s − 4.56·16-s + 1.08·17-s + 0.488·18-s − 2.22·19-s − 8.56·22-s + 1.25·23-s + 1.08·24-s − 12.3·26-s − 5.39·27-s − 3.59·29-s − 2.59·31-s + 7.42·32-s + 7.42·33-s − 2.08·34-s − 0.423·36-s + 7.16·37-s + 4.25·38-s + 10.7·39-s + ⋯
L(s)  = 1  − 1.35·2-s + 0.956·3-s + 0.828·4-s − 1.29·6-s + 0.232·8-s − 0.0852·9-s + 1.35·11-s + 0.792·12-s + 1.79·13-s − 1.14·16-s + 0.263·17-s + 0.115·18-s − 0.510·19-s − 1.82·22-s + 0.261·23-s + 0.222·24-s − 2.43·26-s − 1.03·27-s − 0.668·29-s − 0.466·31-s + 1.31·32-s + 1.29·33-s − 0.356·34-s − 0.0705·36-s + 1.17·37-s + 0.690·38-s + 1.71·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(9.78167\)
Root analytic conductor: \(3.12756\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1225,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.261105209\)
\(L(\frac12)\) \(\approx\) \(1.261105209\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
good2 \( 1 + 1.91T + 2T^{2} \)
3 \( 1 - 1.65T + 3T^{2} \)
11 \( 1 - 4.48T + 11T^{2} \)
13 \( 1 - 6.48T + 13T^{2} \)
17 \( 1 - 1.08T + 17T^{2} \)
19 \( 1 + 2.22T + 19T^{2} \)
23 \( 1 - 1.25T + 23T^{2} \)
29 \( 1 + 3.59T + 29T^{2} \)
31 \( 1 + 2.59T + 31T^{2} \)
37 \( 1 - 7.16T + 37T^{2} \)
41 \( 1 + 3.88T + 41T^{2} \)
43 \( 1 - 3.08T + 43T^{2} \)
47 \( 1 - 6.43T + 47T^{2} \)
53 \( 1 + 3.05T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 + 2.59T + 61T^{2} \)
67 \( 1 - 5.31T + 67T^{2} \)
71 \( 1 - 4.19T + 71T^{2} \)
73 \( 1 + 6.62T + 73T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 - 9.59T + 83T^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 - 4.67T + 97T^{2} \)
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

−9.291038613946797048489790364984, −8.962075992499942653223319947683, −8.383341620945627530388290523659, −7.64675696514773221316472872669, −6.68446372249811372107098263099, −5.83866747426827904559376333103, −4.18830331520501419060002631809, −3.46169801972633001087214736367, −2.06712828106971264844436686403, −1.05583869135541896334512515144, 1.05583869135541896334512515144, 2.06712828106971264844436686403, 3.46169801972633001087214736367, 4.18830331520501419060002631809, 5.83866747426827904559376333103, 6.68446372249811372107098263099, 7.64675696514773221316472872669, 8.383341620945627530388290523659, 8.962075992499942653223319947683, 9.291038613946797048489790364984

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