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Properties

Label 2-136e2-1.1-c1-0-12
Degree $2$
Conductor $18496$
Sign $1$
Analytic cond. $147.691$
Root an. cond. $12.1528$
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  + 4·5-s − 3·9-s + 6·13-s + 11·25-s + 4·29-s + 12·37-s − 8·41-s − 12·45-s − 7·49-s + 14·53-s + 12·61-s + 24·65-s − 16·73-s + 9·81-s − 10·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 18·117-s + ⋯
L(s)  = 1  + 1.78·5-s − 9-s + 1.66·13-s + 11/5·25-s + 0.742·29-s + 1.97·37-s − 1.24·41-s − 1.78·45-s − 49-s + 1.92·53-s + 1.53·61-s + 2.97·65-s − 1.87·73-s + 81-s − 1.05·89-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.66·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18496\)    =    \(2^{6} \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(147.691\)
Root analytic conductor: \(12.1528\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 18496,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.652371820\)
\(L(\frac12)\) \(\approx\) \(3.652371820\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 16 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−15.98848032631728, −14.97488816732378, −14.63922687105297, −13.99816121859749, −13.55402300858703, −13.22924628061914, −12.70275144358781, −11.70922276108041, −11.35440369602896, −10.67986619101645, −10.13127118635891, −9.676933404867136, −8.835922738521738, −8.674548822662833, −7.989127071319161, −6.913484484203853, −6.362625914524769, −5.897043540725122, −5.510044402933424, −4.747875554362473, −3.810328512096064, −2.973715066606335, −2.424216256791042, −1.571943324477696, −0.8436764756464404, 0.8436764756464404, 1.571943324477696, 2.424216256791042, 2.973715066606335, 3.810328512096064, 4.747875554362473, 5.510044402933424, 5.897043540725122, 6.362625914524769, 6.913484484203853, 7.989127071319161, 8.674548822662833, 8.835922738521738, 9.676933404867136, 10.13127118635891, 10.67986619101645, 11.35440369602896, 11.70922276108041, 12.70275144358781, 13.22924628061914, 13.55402300858703, 13.99816121859749, 14.63922687105297, 14.97488816732378, 15.98848032631728

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