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Properties

Label 2-1248-13.3-c1-0-1
Degree $2$
Conductor $1248$
Sign $-0.993 - 0.116i$
Analytic cond. $9.96533$
Root an. cond. $3.15679$
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  + (0.5 + 0.866i)3-s − 2.56·5-s + (0.219 − 0.379i)7-s + (−0.499 + 0.866i)9-s + (1 + 1.73i)11-s + (0.5 + 3.57i)13-s + (−1.28 − 2.21i)15-s + (0.280 − 0.486i)17-s + (−0.561 + 0.972i)19-s + 0.438·21-s + (−2.56 − 4.43i)23-s + 1.56·25-s − 0.999·27-s + (−2.71 − 4.70i)29-s − 8.68·31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s − 1.14·5-s + (0.0828 − 0.143i)7-s + (−0.166 + 0.288i)9-s + (0.301 + 0.522i)11-s + (0.138 + 0.990i)13-s + (−0.330 − 0.572i)15-s + (0.0680 − 0.117i)17-s + (−0.128 + 0.223i)19-s + 0.0956·21-s + (−0.534 − 0.925i)23-s + 0.312·25-s − 0.192·27-s + (−0.504 − 0.874i)29-s − 1.55·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1248\)    =    \(2^{5} \cdot 3 \cdot 13\)
Sign: $-0.993 - 0.116i$
Analytic conductor: \(9.96533\)
Root analytic conductor: \(3.15679\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1248} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1248,\ (\ :1/2),\ -0.993 - 0.116i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5019791539\)
\(L(\frac12)\) \(\approx\) \(0.5019791539\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 - 3.57i)T \)
good5 \( 1 + 2.56T + 5T^{2} \)
7 \( 1 + (-0.219 + 0.379i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.280 + 0.486i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.561 - 0.972i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.56 + 4.43i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.71 + 4.70i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.68T + 31T^{2} \)
37 \( 1 + (-2.28 - 3.95i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.28 + 2.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.78 - 6.54i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 + 2.56T + 53T^{2} \)
59 \( 1 + (5.12 - 8.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.93 - 5.08i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.78 + 3.08i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.561 + 0.972i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 6.12T + 73T^{2} \)
79 \( 1 + 11.8T + 79T^{2} \)
83 \( 1 + 8.24T + 83T^{2} \)
89 \( 1 + (0.123 + 0.213i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.78 - 3.08i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.979750312353033527455004517573, −9.294606357074437108494467701186, −8.433383215805374223185191641354, −7.73349800576297564234370917674, −6.96264265913683572612240278835, −5.94949141454711808145175720344, −4.54747670145220659170177149446, −4.20591705899659565131167298727, −3.22144227727043388868327116709, −1.82250395883263403215168779258, 0.19943707274306792811425025211, 1.74890429060393229444781550900, 3.29713629071724195185934464338, 3.72810394727124743937615187666, 5.10872863829497945049402618717, 5.97568682716403329538305764059, 7.07488795204479524059709318742, 7.72697363852522448249308987697, 8.355656412176695097781602823176, 9.088144461427623789144902943428

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