Properties

Label 2-7e2-1.1-c3-0-2
Degree $2$
Conductor $49$
Sign $-1$
Analytic cond. $2.89109$
Root an. cond. $1.70032$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5·2-s + 17·4-s − 45·8-s − 27·9-s − 68·11-s + 89·16-s + 135·18-s + 340·22-s − 40·23-s − 125·25-s − 166·29-s − 85·32-s − 459·36-s + 450·37-s − 180·43-s − 1.15e3·44-s + 200·46-s + 625·50-s + 590·53-s + 830·58-s − 287·64-s − 740·67-s + 688·71-s + 1.21e3·72-s − 2.25e3·74-s − 1.38e3·79-s + 729·81-s + ⋯
L(s)  = 1  − 1.76·2-s + 17/8·4-s − 1.98·8-s − 9-s − 1.86·11-s + 1.39·16-s + 1.76·18-s + 3.29·22-s − 0.362·23-s − 25-s − 1.06·29-s − 0.469·32-s − 2.12·36-s + 1.99·37-s − 0.638·43-s − 3.96·44-s + 0.641·46-s + 1.76·50-s + 1.52·53-s + 1.87·58-s − 0.560·64-s − 1.34·67-s + 1.15·71-s + 1.98·72-s − 3.53·74-s − 1.97·79-s + 81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(49\)    =    \(7^{2}\)
Sign: $-1$
Analytic conductor: \(2.89109\)
Root analytic conductor: \(1.70032\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{49} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 49,\ (\ :3/2),\ -1)\)

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)$
bad7 \( 1 \)
good2 \( 1 + 5 T + p^{3} T^{2} \)
3 \( 1 + p^{3} T^{2} \)
5 \( 1 + p^{3} T^{2} \)
11 \( 1 + 68 T + p^{3} T^{2} \)
13 \( 1 + p^{3} T^{2} \)
17 \( 1 + p^{3} T^{2} \)
19 \( 1 + p^{3} T^{2} \)
23 \( 1 + 40 T + p^{3} T^{2} \)
29 \( 1 + 166 T + p^{3} T^{2} \)
31 \( 1 + p^{3} T^{2} \)
37 \( 1 - 450 T + p^{3} T^{2} \)
41 \( 1 + p^{3} T^{2} \)
43 \( 1 + 180 T + p^{3} T^{2} \)
47 \( 1 + p^{3} T^{2} \)
53 \( 1 - 590 T + p^{3} T^{2} \)
59 \( 1 + p^{3} T^{2} \)
61 \( 1 + p^{3} T^{2} \)
67 \( 1 + 740 T + p^{3} T^{2} \)
71 \( 1 - 688 T + p^{3} T^{2} \)
73 \( 1 + p^{3} T^{2} \)
79 \( 1 + 1384 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + p^{3} T^{2} \)
97 \( 1 + p^{3} T^{2} \)
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

−15.02894184714965737715717038841, −13.30442063818121698660187280692, −11.64156989511880964719127413578, −10.69837535176370904724193150519, −9.657416614835928522773169291177, −8.353921336728576853261425161019, −7.56866168292611020750437653852, −5.77862461637823030407687672667, −2.50258280312093899836671041842, 0, 2.50258280312093899836671041842, 5.77862461637823030407687672667, 7.56866168292611020750437653852, 8.353921336728576853261425161019, 9.657416614835928522773169291177, 10.69837535176370904724193150519, 11.64156989511880964719127413578, 13.30442063818121698660187280692, 15.02894184714965737715717038841

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