Properties

Label 2-1216-8.5-c1-0-28
Degree $2$
Conductor $1216$
Sign $-0.258 + 0.965i$
Analytic cond. $9.70980$
Root an. cond. $3.11605$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.04i·5-s − 0.418·7-s + 3·9-s − 5.27i·11-s + 2.62i·13-s + 3.27·17-s i·19-s − 3.46·23-s − 4.27·25-s + 2.62i·29-s − 6.09·31-s + 1.27i·35-s − 9.55i·37-s + 4.54·41-s − 2.72i·43-s + ⋯
L(s)  = 1  − 1.36i·5-s − 0.158·7-s + 9-s − 1.59i·11-s + 0.728i·13-s + 0.794·17-s − 0.229i·19-s − 0.722·23-s − 0.854·25-s + 0.487i·29-s − 1.09·31-s + 0.215i·35-s − 1.57i·37-s + 0.710·41-s − 0.415i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.258 + 0.965i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.258 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.550361323\)
\(L(\frac12)\) \(\approx\) \(1.550361323\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good3 \( 1 - 3T^{2} \)
5 \( 1 + 3.04iT - 5T^{2} \)
7 \( 1 + 0.418T + 7T^{2} \)
11 \( 1 + 5.27iT - 11T^{2} \)
13 \( 1 - 2.62iT - 13T^{2} \)
17 \( 1 - 3.27T + 17T^{2} \)
23 \( 1 + 3.46T + 23T^{2} \)
29 \( 1 - 2.62iT - 29T^{2} \)
31 \( 1 + 6.09T + 31T^{2} \)
37 \( 1 + 9.55iT - 37T^{2} \)
41 \( 1 - 4.54T + 41T^{2} \)
43 \( 1 + 2.72iT - 43T^{2} \)
47 \( 1 - 0.418T + 47T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 - 6.54iT - 59T^{2} \)
61 \( 1 + 3.04iT - 61T^{2} \)
67 \( 1 - 6.54iT - 67T^{2} \)
71 \( 1 - 11.3T + 71T^{2} \)
73 \( 1 + 3.27T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 17.0iT - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 - 16.5T + 97T^{2} \)
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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

−9.300117005932263488742384645401, −8.807849970460476445880146326983, −7.981639441508249745553963450590, −7.09177050017374049593318109586, −5.96514610925549358865688515524, −5.28208152859081322757927669281, −4.26495244049795692351410865526, −3.48783627673262881550463045916, −1.80386150065715661652496502166, −0.68827073265550031454047029351, 1.67657201673385603884662542277, 2.80040268024149812142606011721, 3.79522109804007823105823398979, 4.75296332021822054274695488414, 5.94565955816806508714225882312, 6.81404556620794821987041429226, 7.42605528874647181653680316391, 8.017362304981690183102662729671, 9.611321305326071882233976309459, 9.942334024131057043875128936318

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