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Properties

Label 2-352-11.4-c1-0-7
Degree $2$
Conductor $352$
Sign $0.820 + 0.572i$
Analytic cond. $2.81073$
Root an. cond. $1.67652$
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  + (−0.385 + 1.18i)3-s + (3.03 − 2.20i)5-s + (−1.04 − 3.21i)7-s + (1.17 + 0.850i)9-s + (−1.01 − 3.15i)11-s + (−3.68 − 2.67i)13-s + (1.44 + 4.45i)15-s + (0.239 − 0.174i)17-s + (−0.887 + 2.73i)19-s + 4.21·21-s + 8.74·23-s + (2.81 − 8.66i)25-s + (−4.48 + 3.25i)27-s + (1.21 + 3.73i)29-s + (4.68 + 3.40i)31-s + ⋯
L(s)  = 1  + (−0.222 + 0.684i)3-s + (1.35 − 0.987i)5-s + (−0.395 − 1.21i)7-s + (0.390 + 0.283i)9-s + (−0.304 − 0.952i)11-s + (−1.02 − 0.742i)13-s + (0.373 + 1.14i)15-s + (0.0581 − 0.0422i)17-s + (−0.203 + 0.626i)19-s + 0.920·21-s + 1.82·23-s + (0.563 − 1.73i)25-s + (−0.862 + 0.626i)27-s + (0.225 + 0.693i)29-s + (0.840 + 0.610i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.820 + 0.572i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.820 + 0.572i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39204 - 0.437560i\)
\(L(\frac12)\) \(\approx\) \(1.39204 - 0.437560i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (1.01 + 3.15i)T \)
good3 \( 1 + (0.385 - 1.18i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-3.03 + 2.20i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.04 + 3.21i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.68 + 2.67i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.239 + 0.174i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.887 - 2.73i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.74T + 23T^{2} \)
29 \( 1 + (-1.21 - 3.73i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.68 - 3.40i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.784 - 2.41i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.23 - 6.87i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.03T + 43T^{2} \)
47 \( 1 + (1.57 - 4.83i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.78 + 7.10i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.216 - 0.666i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.27 - 1.65i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 5.78T + 67T^{2} \)
71 \( 1 + (1.94 - 1.41i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.89 - 5.84i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (0.502 + 0.365i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-8.76 + 6.37i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 2.37T + 89T^{2} \)
97 \( 1 + (-8.00 - 5.81i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.02570658269621042197094967098, −10.23834376054751110686538097947, −9.836362405701199333670786986616, −8.827961088800177658869101522684, −7.64203656998875927101727458660, −6.40857552899802571263598989841, −5.22343147472141763650726674652, −4.68408776071163450682371434735, −3.09844808718354313466842862094, −1.14658022913607365671930897264, 2.03596667791277223989514824221, 2.71197188964547536483146525595, 4.84987200540593452315749488693, 5.99748094717133380382190927452, 6.72690225448967008097260904930, 7.38343451483633651685088332776, 9.192837357800754857896139854053, 9.580671670907711969218098481370, 10.54475554353675972134073284141, 11.73848666420293262970702616818

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