Properties

Label 2-1216-8.5-c1-0-2
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  + 2i·3-s − 3.31i·5-s − 3.31·7-s − 9-s + 5i·11-s + 6.63·15-s + 5·17-s i·19-s − 6.63i·21-s − 6.63·23-s − 6·25-s + 4i·27-s + 6.63i·29-s − 10·33-s + 11i·35-s + ⋯
L(s)  = 1  + 1.15i·3-s − 1.48i·5-s − 1.25·7-s − 0.333·9-s + 1.50i·11-s + 1.71·15-s + 1.21·17-s − 0.229i·19-s − 1.44i·21-s − 1.38·23-s − 1.20·25-s + 0.769i·27-s + 1.23i·29-s − 1.74·33-s + 1.85i·35-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

\(L(1)\) \(\approx\) \(0.8714690872\)
\(L(\frac12)\) \(\approx\) \(0.8714690872\)
\(L(\frac{3}{2})\) 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 \( 1 - 2iT - 3T^{2} \)
5 \( 1 + 3.31iT - 5T^{2} \)
7 \( 1 + 3.31T + 7T^{2} \)
11 \( 1 - 5iT - 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
23 \( 1 + 6.63T + 23T^{2} \)
29 \( 1 - 6.63iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6.63iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + 9.94T + 47T^{2} \)
53 \( 1 - 13.2iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 - 9.94iT - 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 - 6.63T + 71T^{2} \)
73 \( 1 + 9T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + 4T + 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−9.951388674229985195346896363450, −9.433646855425412035910706237909, −8.708425152823158241460989971443, −7.69675196374886512396543045766, −6.67451998530025654023698139397, −5.52161813586427681127493154043, −4.82318302801540567901277684336, −4.12950482803184405001146979328, −3.19805394882484495721313956110, −1.50571303898887487008178547615, 0.37043274595792950317024668851, 2.08225422170575501897262664898, 3.17598049585244861117592391403, 3.65612567421964293048161167803, 5.73709645719161143310994940025, 6.29917760001774489458982200150, 6.76524858234280497576853962935, 7.73667585273129870283555871511, 8.248346288507496600069670339839, 9.717674508119770023477502100476

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