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Properties

Label 2-1216-152.45-c1-0-10
Degree $2$
Conductor $1216$
Sign $-0.842 - 0.538i$
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  + (−2.17 + 1.25i)3-s + (−2.95 + 1.70i)5-s + 2.54·7-s + (1.66 − 2.87i)9-s + 1.25i·11-s + (4.22 + 2.44i)13-s + (4.29 − 7.43i)15-s + (3.99 + 6.91i)17-s + (3.96 − 1.81i)19-s + (−5.54 + 3.19i)21-s + (0.355 − 0.615i)23-s + (3.32 − 5.75i)25-s + 0.806i·27-s + (2.45 + 1.41i)29-s − 5.32·31-s + ⋯
L(s)  = 1  + (−1.25 + 0.725i)3-s + (−1.32 + 0.763i)5-s + 0.961·7-s + (0.553 − 0.958i)9-s + 0.378i·11-s + (1.17 + 0.677i)13-s + (1.10 − 1.91i)15-s + (0.968 + 1.67i)17-s + (0.908 − 0.416i)19-s + (−1.20 + 0.698i)21-s + (0.0741 − 0.128i)23-s + (0.664 − 1.15i)25-s + 0.155i·27-s + (0.455 + 0.262i)29-s − 0.956·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.842 - 0.538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1216\)    =    \(2^{6} \cdot 19\)
Sign: $-0.842 - 0.538i$
Analytic conductor: \(9.70980\)
Root analytic conductor: \(3.11605\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1216} (1185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1216,\ (\ :1/2),\ -0.842 - 0.538i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8576598715\)
\(L(\frac12)\) \(\approx\) \(0.8576598715\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.96 + 1.81i)T \)
good3 \( 1 + (2.17 - 1.25i)T + (1.5 - 2.59i)T^{2} \)
5 \( 1 + (2.95 - 1.70i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 - 2.54T + 7T^{2} \)
11 \( 1 - 1.25iT - 11T^{2} \)
13 \( 1 + (-4.22 - 2.44i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.99 - 6.91i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.355 + 0.615i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.45 - 1.41i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.32T + 31T^{2} \)
37 \( 1 - 3.20iT - 37T^{2} \)
41 \( 1 + (-5.26 - 9.11i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.69 + 3.29i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.00 - 6.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.905 + 0.522i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.46 + 3.73i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.9 + 6.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.72 + 1.57i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.34 + 10.9i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.03 - 5.25i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.17 + 2.04i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.2iT - 83T^{2} \)
89 \( 1 + (-1.39 + 2.40i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.77 + 9.99i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.47682531819469828983747794322, −9.399110504704246101221164687422, −8.208419919690045652421019690604, −7.69690158540089337773225874461, −6.61277415753897494003841621980, −5.87349706712213066560094289584, −4.81857667636349808675599696267, −4.13426579295482231074458771049, −3.34757582387841310197949757653, −1.34173864118242339708699581936, 0.56768066431474443592116803171, 1.24933751427778644507899603290, 3.28583318723660258907280094761, 4.37591429606601213772053714798, 5.40639113010942040065815081429, 5.68921863433073224562754215565, 7.18102583457373954856102890573, 7.62363302943251319107111965452, 8.335111167045111382874667102688, 9.239400290659438640033551960918

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