Properties

Label 2-1216-76.75-c1-0-19
Degree $2$
Conductor $1216$
Sign $0.606 - 0.794i$
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  + 2.64·3-s + 1.73i·7-s + 4.00·9-s + 3.46i·11-s + 4.58i·13-s + 3·17-s + (−2.64 + 3.46i)19-s + 4.58i·21-s − 5.19i·23-s − 5·25-s + 2.64·27-s + 4.58i·29-s + 5.29·31-s + 9.16i·33-s − 9.16i·37-s + ⋯
L(s)  = 1  + 1.52·3-s + 0.654i·7-s + 1.33·9-s + 1.04i·11-s + 1.27i·13-s + 0.727·17-s + (−0.606 + 0.794i)19-s + 0.999i·21-s − 1.08i·23-s − 25-s + 0.509·27-s + 0.850i·29-s + 0.950·31-s + 1.59i·33-s − 1.50i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.606 - 0.794i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.606 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.719071980\)
\(L(\frac12)\) \(\approx\) \(2.719071980\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 3.46i)T \)
good3 \( 1 - 2.64T + 3T^{2} \)
5 \( 1 + 5T^{2} \)
7 \( 1 - 1.73iT - 7T^{2} \)
11 \( 1 - 3.46iT - 11T^{2} \)
13 \( 1 - 4.58iT - 13T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
23 \( 1 + 5.19iT - 23T^{2} \)
29 \( 1 - 4.58iT - 29T^{2} \)
31 \( 1 - 5.29T + 31T^{2} \)
37 \( 1 + 9.16iT - 37T^{2} \)
41 \( 1 + 9.16iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 4.58iT - 53T^{2} \)
59 \( 1 - 7.93T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 - 2.64T + 67T^{2} \)
71 \( 1 - 15.8T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + 5.29T + 79T^{2} \)
83 \( 1 - 6.92iT - 83T^{2} \)
89 \( 1 + 18.3iT - 89T^{2} \)
97 \( 1 + 9.16iT - 97T^{2} \)
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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

−9.706541227280144623280818654583, −8.902190109779649140008674682521, −8.430780970831348136890523216815, −7.51195481040265438388358110732, −6.79531537039988908275275926686, −5.63848811164763486835835423818, −4.36818341390879373943825858462, −3.71348254690985208809493685977, −2.40196928220026559881875904314, −1.89671423949553437373992045806, 1.01191301252709051843371456805, 2.56343857070413168442949455301, 3.30717386131226925748647439109, 4.05305044843527508616730846062, 5.33899556398969600742191701030, 6.38102699494528626698854652711, 7.52615667480879562802527628363, 8.094935916779749550301057410074, 8.551147143636774919400130716577, 9.700086963599393867248117617565

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