Properties

Label 2-1216-152.75-c1-0-10
Degree $2$
Conductor $1216$
Sign $0.380 - 0.924i$
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  + 2.64i·3-s − 3.46i·5-s + i·7-s − 4.00·9-s + 2.64·13-s + 9.16·15-s − 3·17-s + (3.46 + 2.64i)19-s − 2.64·21-s + 3i·23-s − 6.99·25-s − 2.64i·27-s + 7.93·29-s + 3.46·35-s + 10.5·37-s + ⋯
L(s)  = 1  + 1.52i·3-s − 1.54i·5-s + 0.377i·7-s − 1.33·9-s + 0.733·13-s + 2.36·15-s − 0.727·17-s + (0.794 + 0.606i)19-s − 0.577·21-s + 0.625i·23-s − 1.39·25-s − 0.509i·27-s + 1.47·29-s + 0.585·35-s + 1.73·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.380 - 0.924i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.380 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.642590216\)
\(L(\frac12)\) \(\approx\) \(1.642590216\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.46 - 2.64i)T \)
good3 \( 1 - 2.64iT - 3T^{2} \)
5 \( 1 + 3.46iT - 5T^{2} \)
7 \( 1 - iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10.5T + 37T^{2} \)
41 \( 1 - 9.16iT - 41T^{2} \)
43 \( 1 - 10.3T + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 7.93T + 53T^{2} \)
59 \( 1 - 7.93iT - 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + 13.2iT - 67T^{2} \)
71 \( 1 + 9.16T + 71T^{2} \)
73 \( 1 - 7T + 73T^{2} \)
79 \( 1 - 9.16T + 79T^{2} \)
83 \( 1 + 17.3T + 83T^{2} \)
89 \( 1 + 9.16iT - 89T^{2} \)
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

−9.618877854167099802547885963585, −9.223090256474745727695395940491, −8.553991463566915541061541345570, −7.80055973652994957813760024687, −6.14260572607316727443854671505, −5.46093461477370256724322409018, −4.58213200956875764198309959590, −4.14718359850822826139172973262, −2.92841693220948713389994166421, −1.17975443599525605456101406317, 0.857626994542693978182521476933, 2.30826587226868188248152410530, 2.96014391000346743989950961517, 4.23250166507380627472293954711, 5.81489488721479525954980117171, 6.56377568558692546066515383768, 6.98323288894108241213629649794, 7.68689313001762499627289724435, 8.485765015104667628044036504200, 9.600546859870100895184366361389

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