Properties

Label 2-1216-8.5-c1-0-13
Degree $2$
Conductor $1216$
Sign $0.707 - 0.707i$
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  + i·3-s + 3·7-s + 2·9-s + 3i·13-s + 3·17-s i·19-s + 3i·21-s − 9·23-s + 5·25-s + 5i·27-s − 9i·29-s + 6·31-s − 6i·37-s − 3·39-s + 6·41-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.13·7-s + 0.666·9-s + 0.832i·13-s + 0.727·17-s − 0.229i·19-s + 0.654i·21-s − 1.87·23-s + 25-s + 0.962i·27-s − 1.67i·29-s + 1.07·31-s − 0.986i·37-s − 0.480·39-s + 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.707 - 0.707i$
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: \((2,\ 1216,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.024292831\)
\(L(\frac12)\) \(\approx\) \(2.024292831\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 3iT - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
23 \( 1 + 9T + 23T^{2} \)
29 \( 1 + 9iT - 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 - 3iT - 59T^{2} \)
61 \( 1 - 6iT - 61T^{2} \)
67 \( 1 - 5iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−9.862685587425394053191256341060, −9.145816530441427860233828752262, −8.105497545734099352261462033095, −7.58651326697501713051468396478, −6.46359765609492926333934285601, −5.51248430504078833539322710571, −4.39078225807291656867764951514, −4.14829853802996598658054914664, −2.52003549943530161976396997584, −1.32624340253520185142858625608, 1.07113267627942141792377710412, 2.04295540264713895441864435695, 3.40579877635572868078992330615, 4.57082014948238083712548335703, 5.35132135944355142848853256403, 6.37254600139472960295505130365, 7.28668043003824428231687611588, 8.040319249155155768203095438866, 8.462742243307877688021409433914, 9.809603195376979946669488949759

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