Properties

Label 2-35e2-1.1-c3-0-122
Degree $2$
Conductor $1225$
Sign $-1$
Analytic cond. $72.2773$
Root an. cond. $8.50160$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2·3-s − 7·4-s − 2·6-s − 15·8-s − 23·9-s − 8·11-s + 14·12-s + 28·13-s + 41·16-s + 54·17-s − 23·18-s + 110·19-s − 8·22-s − 48·23-s + 30·24-s + 28·26-s + 100·27-s − 110·29-s − 12·31-s + 161·32-s + 16·33-s + 54·34-s + 161·36-s + 246·37-s + 110·38-s − 56·39-s + ⋯
L(s)  = 1  + 0.353·2-s − 0.384·3-s − 7/8·4-s − 0.136·6-s − 0.662·8-s − 0.851·9-s − 0.219·11-s + 0.336·12-s + 0.597·13-s + 0.640·16-s + 0.770·17-s − 0.301·18-s + 1.32·19-s − 0.0775·22-s − 0.435·23-s + 0.255·24-s + 0.211·26-s + 0.712·27-s − 0.704·29-s − 0.0695·31-s + 0.889·32-s + 0.0844·33-s + 0.272·34-s + 0.745·36-s + 1.09·37-s + 0.469·38-s − 0.229·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1225\)    =    \(5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(72.2773\)
Root analytic conductor: \(8.50160\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1225} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1225,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 - T + p^{3} T^{2} \)
3 \( 1 + 2 T + p^{3} T^{2} \)
11 \( 1 + 8 T + p^{3} T^{2} \)
13 \( 1 - 28 T + p^{3} T^{2} \)
17 \( 1 - 54 T + p^{3} T^{2} \)
19 \( 1 - 110 T + p^{3} T^{2} \)
23 \( 1 + 48 T + p^{3} T^{2} \)
29 \( 1 + 110 T + p^{3} T^{2} \)
31 \( 1 + 12 T + p^{3} T^{2} \)
37 \( 1 - 246 T + p^{3} T^{2} \)
41 \( 1 + 182 T + p^{3} T^{2} \)
43 \( 1 + 128 T + p^{3} T^{2} \)
47 \( 1 - 324 T + p^{3} T^{2} \)
53 \( 1 - 162 T + p^{3} T^{2} \)
59 \( 1 + 810 T + p^{3} T^{2} \)
61 \( 1 - 8 p T + p^{3} T^{2} \)
67 \( 1 + 244 T + p^{3} T^{2} \)
71 \( 1 + 768 T + p^{3} T^{2} \)
73 \( 1 + 702 T + p^{3} T^{2} \)
79 \( 1 - 440 T + p^{3} T^{2} \)
83 \( 1 + 1302 T + p^{3} T^{2} \)
89 \( 1 + 730 T + p^{3} T^{2} \)
97 \( 1 - 294 T + p^{3} T^{2} \)
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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

−8.953384588572920504375840476603, −8.197180197794439656730769342609, −7.37173749140732420786463646080, −5.98195015228847581908013282634, −5.63447974264080229985561161946, −4.74709078617697647876790167589, −3.67309813869389593628344148345, −2.89266512499027057938802213026, −1.15098830882669493082327319620, 0, 1.15098830882669493082327319620, 2.89266512499027057938802213026, 3.67309813869389593628344148345, 4.74709078617697647876790167589, 5.63447974264080229985561161946, 5.98195015228847581908013282634, 7.37173749140732420786463646080, 8.197180197794439656730769342609, 8.953384588572920504375840476603

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