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Properties

Label 2-1148-7.2-c1-0-4
Degree $2$
Conductor $1148$
Sign $-0.653 - 0.757i$
Analytic cond. $9.16682$
Root an. cond. $3.02767$
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  + (0.864 + 1.49i)3-s + (0.0691 − 0.119i)5-s + (0.329 + 2.62i)7-s + (0.00362 − 0.00628i)9-s + (−1.34 − 2.33i)11-s − 5.05·13-s + 0.239·15-s + (2.19 + 3.79i)17-s + (−2.86 + 4.96i)19-s + (−3.64 + 2.76i)21-s + (−4.04 + 7.01i)23-s + (2.49 + 4.31i)25-s + 5.20·27-s + 3.49·29-s + (1.42 + 2.46i)31-s + ⋯
L(s)  = 1  + (0.499 + 0.864i)3-s + (0.0309 − 0.0535i)5-s + (0.124 + 0.992i)7-s + (0.00120 − 0.00209i)9-s + (−0.406 − 0.703i)11-s − 1.40·13-s + 0.0617·15-s + (0.531 + 0.920i)17-s + (−0.657 + 1.13i)19-s + (−0.796 + 0.603i)21-s + (−0.844 + 1.46i)23-s + (0.498 + 0.862i)25-s + 1.00·27-s + 0.648·29-s + (0.255 + 0.442i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1148\)    =    \(2^{2} \cdot 7 \cdot 41\)
Sign: $-0.653 - 0.757i$
Analytic conductor: \(9.16682\)
Root analytic conductor: \(3.02767\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1148} (821, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1148,\ (\ :1/2),\ -0.653 - 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.487537930\)
\(L(\frac12)\) \(\approx\) \(1.487537930\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.329 - 2.62i)T \)
41 \( 1 + T \)
good3 \( 1 + (-0.864 - 1.49i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.0691 + 0.119i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.34 + 2.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.05T + 13T^{2} \)
17 \( 1 + (-2.19 - 3.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.86 - 4.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.04 - 7.01i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.49T + 29T^{2} \)
31 \( 1 + (-1.42 - 2.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.02 + 3.50i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (0.274 - 0.476i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.93 + 10.2i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.92 - 8.53i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.54 + 6.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.40 - 2.42i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.5T + 71T^{2} \)
73 \( 1 + (-3.90 - 6.76i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.26 + 9.12i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.44T + 83T^{2} \)
89 \( 1 + (0.892 - 1.54i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 5.82T + 97T^{2} \)
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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

−9.873150697555116604626345742185, −9.484129508255689315685778611835, −8.380335896255668632466091557963, −8.053135739233892775572824064918, −6.70981607273861381471293062552, −5.64363424204487944712971838307, −5.03736142053016579325195075408, −3.82068847912794007369554512121, −3.08126750357804505767231641256, −1.85434210462791323377878109062, 0.58180417142751584032936493580, 2.16638002910002686062249546012, 2.81548990506226174720265343918, 4.53305177404056873322248268143, 4.86778539766422466510476665034, 6.69039421436168556482978193654, 6.93358742004422194894127798541, 7.87582472682557294850107643158, 8.333579325250933402342761499048, 9.700749815218497726501791259551

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