Properties

Label 2-1216-1.1-c1-0-25
Degree $2$
Conductor $1216$
Sign $-1$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.56·3-s + 2.56·5-s − 3·7-s − 0.561·9-s + 0.561·11-s + 1.56·13-s − 4·15-s + 0.123·17-s + 19-s + 4.68·21-s − 5.56·23-s + 1.56·25-s + 5.56·27-s − 4.68·29-s − 9.12·31-s − 0.876·33-s − 7.68·35-s − 7.12·37-s − 2.43·39-s + 4·41-s + 5.43·43-s − 1.43·45-s + 3.68·47-s + 2·49-s − 0.192·51-s + 4.43·53-s + 1.43·55-s + ⋯
L(s)  = 1  − 0.901·3-s + 1.14·5-s − 1.13·7-s − 0.187·9-s + 0.169·11-s + 0.433·13-s − 1.03·15-s + 0.0298·17-s + 0.229·19-s + 1.02·21-s − 1.15·23-s + 0.312·25-s + 1.07·27-s − 0.869·29-s − 1.63·31-s − 0.152·33-s − 1.29·35-s − 1.17·37-s − 0.390·39-s + 0.624·41-s + 0.829·43-s − 0.214·45-s + 0.537·47-s + 0.285·49-s − 0.0269·51-s + 0.609·53-s + 0.193·55-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: no
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 \( 1 + 1.56T + 3T^{2} \)
5 \( 1 - 2.56T + 5T^{2} \)
7 \( 1 + 3T + 7T^{2} \)
11 \( 1 - 0.561T + 11T^{2} \)
13 \( 1 - 1.56T + 13T^{2} \)
17 \( 1 - 0.123T + 17T^{2} \)
23 \( 1 + 5.56T + 23T^{2} \)
29 \( 1 + 4.68T + 29T^{2} \)
31 \( 1 + 9.12T + 31T^{2} \)
37 \( 1 + 7.12T + 37T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 5.43T + 43T^{2} \)
47 \( 1 - 3.68T + 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 + 4.43T + 59T^{2} \)
61 \( 1 + 14.8T + 61T^{2} \)
67 \( 1 + 0.438T + 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + 9.24T + 73T^{2} \)
79 \( 1 + 10T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 + 15.3T + 89T^{2} \)
97 \( 1 - 7.12T + 97T^{2} \)
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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

−9.401534569465140448478223180627, −8.820604068959892843241852593888, −7.45554932226532611543732923011, −6.55352104764641269045717217135, −5.74448315765810847195685347969, −5.62603650770663543413790010552, −4.10152857220081832419368412288, −2.98956207569121307769104942490, −1.70831221281079586954224147811, 0, 1.70831221281079586954224147811, 2.98956207569121307769104942490, 4.10152857220081832419368412288, 5.62603650770663543413790010552, 5.74448315765810847195685347969, 6.55352104764641269045717217135, 7.45554932226532611543732923011, 8.820604068959892843241852593888, 9.401534569465140448478223180627

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