Properties

Label 2-1216-1.1-c1-0-32
Degree $2$
Conductor $1216$
Sign $-1$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 3·5-s − 7-s + 9-s − 3·11-s + 4·13-s − 6·15-s − 3·17-s − 19-s − 2·21-s + 4·25-s − 4·27-s − 6·29-s − 4·31-s − 6·33-s + 3·35-s − 2·37-s + 8·39-s − 6·41-s + 43-s − 3·45-s − 3·47-s − 6·49-s − 6·51-s − 12·53-s + 9·55-s − 2·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.10·13-s − 1.54·15-s − 0.727·17-s − 0.229·19-s − 0.436·21-s + 4/5·25-s − 0.769·27-s − 1.11·29-s − 0.718·31-s − 1.04·33-s + 0.507·35-s − 0.328·37-s + 1.28·39-s − 0.937·41-s + 0.152·43-s − 0.447·45-s − 0.437·47-s − 6/7·49-s − 0.840·51-s − 1.64·53-s + 1.21·55-s − 0.264·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-1$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1216} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
7 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 12 T + p T^{2} \)
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

−9.035139763634884351001508227661, −8.457332600420497477897703269253, −7.86885586734404145925295723068, −7.15811652768508964807251071197, −6.05813289271022268292942638231, −4.80419327140120764096716079767, −3.67310116806022314895429635564, −3.31935920587958999413340629739, −2.04259190544357015606449170426, 0, 2.04259190544357015606449170426, 3.31935920587958999413340629739, 3.67310116806022314895429635564, 4.80419327140120764096716079767, 6.05813289271022268292942638231, 7.15811652768508964807251071197, 7.86885586734404145925295723068, 8.457332600420497477897703269253, 9.035139763634884351001508227661

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