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Properties

Label 2-352-11.3-c1-0-0
Degree $2$
Conductor $352$
Sign $-0.919 - 0.392i$
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  + (0.945 + 2.90i)3-s + (−2.43 − 1.76i)5-s + (−0.483 + 1.48i)7-s + (−5.14 + 3.73i)9-s + (−3.21 + 0.815i)11-s + (−2.95 + 2.14i)13-s + (2.84 − 8.74i)15-s + (3.62 + 2.63i)17-s + (0.848 + 2.61i)19-s − 4.78·21-s + 4.77·23-s + (1.24 + 3.83i)25-s + (−8.31 − 6.03i)27-s + (1.53 − 4.72i)29-s + (−0.394 + 0.286i)31-s + ⋯
L(s)  = 1  + (0.545 + 1.67i)3-s + (−1.08 − 0.790i)5-s + (−0.182 + 0.562i)7-s + (−1.71 + 1.24i)9-s + (−0.969 + 0.245i)11-s + (−0.820 + 0.596i)13-s + (0.733 − 2.25i)15-s + (0.878 + 0.638i)17-s + (0.194 + 0.599i)19-s − 1.04·21-s + 0.995·23-s + (0.249 + 0.767i)25-s + (−1.59 − 1.16i)27-s + (0.285 − 0.877i)29-s + (−0.0708 + 0.0514i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $-0.919 - 0.392i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ -0.919 - 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185773 + 0.908114i\)
\(L(\frac12)\) \(\approx\) \(0.185773 + 0.908114i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.21 - 0.815i)T \)
good3 \( 1 + (-0.945 - 2.90i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.43 + 1.76i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.483 - 1.48i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.95 - 2.14i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.62 - 2.63i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.848 - 2.61i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.77T + 23T^{2} \)
29 \( 1 + (-1.53 + 4.72i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.394 - 0.286i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.28 - 10.1i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.87 + 11.9i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.45T + 43T^{2} \)
47 \( 1 + (-3.10 - 9.54i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.26 + 3.09i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.76 - 8.49i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.75 + 1.27i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 0.709T + 67T^{2} \)
71 \( 1 + (0.654 + 0.475i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.163 - 0.502i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.70 - 3.41i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.73 - 4.16i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 7.76T + 89T^{2} \)
97 \( 1 + (2.93 - 2.13i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.92016515906435327071233931897, −10.72942442293805259369545520615, −9.982043797460385382268485366982, −9.114619781834648795657080817256, −8.376791018179645007649965376078, −7.56403208915759376304362732933, −5.55347805948240643430389816208, −4.74766686520069704178673566643, −3.92152292168912939128929011580, −2.77445724219536714516702356418, 0.58808577891943064826750689529, 2.65409112385666367686263045580, 3.35387390102636686640935641752, 5.28149470208903270769948538875, 6.77525290834504355681901517322, 7.48730917342013462764212904235, 7.70383171421005629619407561026, 8.899045313641513277680394241720, 10.35164235385366722893354734150, 11.26065473675989049486360364809

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