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Properties

Label 2-1216-152.125-c1-0-19
Degree $2$
Conductor $1216$
Sign $0.163 - 0.986i$
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  + (2.59 + 1.5i)3-s + (2.73 + 1.58i)5-s − 3.16·7-s + (3 + 5.19i)9-s + 3i·11-s + (5.47 − 3.16i)13-s + (4.74 + 8.21i)15-s + (−2 + 3.46i)17-s + (2.59 − 3.5i)19-s + (−8.21 − 4.74i)21-s + (−4.74 − 8.21i)23-s + (2.5 + 4.33i)25-s + 9i·27-s + (−2.73 + 1.58i)29-s − 3.16·31-s + ⋯
L(s)  = 1  + (1.49 + 0.866i)3-s + (1.22 + 0.707i)5-s − 1.19·7-s + (1 + 1.73i)9-s + 0.904i·11-s + (1.51 − 0.877i)13-s + (1.22 + 2.12i)15-s + (−0.485 + 0.840i)17-s + (0.596 − 0.802i)19-s + (−1.79 − 1.03i)21-s + (−0.989 − 1.71i)23-s + (0.5 + 0.866i)25-s + 1.73i·27-s + (−0.508 + 0.293i)29-s − 0.567·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $0.163 - 0.986i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ 0.163 - 0.986i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.115404817\)
\(L(\frac12)\) \(\approx\) \(3.115404817\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.59 + 3.5i)T \)
good3 \( 1 + (-2.59 - 1.5i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.73 - 1.58i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.16T + 7T^{2} \)
11 \( 1 - 3iT - 11T^{2} \)
13 \( 1 + (-5.47 + 3.16i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2 - 3.46i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (4.74 + 8.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.73 - 1.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.16T + 31T^{2} \)
37 \( 1 - 3.16iT - 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.66 - 5i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.58 + 2.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.06 + 3.5i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.73 - 1.58i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.33 - 2.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.16 - 5.47i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7iT - 83T^{2} \)
89 \( 1 + (4 + 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.5 + 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.917320990502161566407805964323, −9.164049581333044800946340916788, −8.616148705078616046364524105538, −7.56738369430431371605215741428, −6.49502528975896506748560144969, −5.90600148588166298620786536054, −4.48861767626762367829684276164, −3.52820327954461375091435918424, −2.82711933683816202635238461747, −1.98379748561611631081146945806, 1.22799881117220384333217070861, 2.06719604266135215207457301960, 3.24629311627295587345124803465, 3.88074248135191052832206492813, 5.76804431749625148227533397785, 6.12651564018351310458923912277, 7.15991840118303760100014274239, 8.029343280991045680949748098008, 9.048949909975405492372236360586, 9.212020817633316940872865604269

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