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Properties

Label 2-352-44.19-c1-0-9
Degree $2$
Conductor $352$
Sign $0.906 + 0.421i$
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  + (2.26 − 0.734i)3-s + (0.0210 + 0.0153i)5-s + (0.140 − 0.432i)7-s + (2.14 − 1.55i)9-s + (3.11 + 1.12i)11-s + (−1.27 − 1.76i)13-s + (0.0589 + 0.0191i)15-s + (1.94 − 2.67i)17-s + (0.802 + 2.47i)19-s − 1.07i·21-s + 3.69i·23-s + (−1.54 − 4.75i)25-s + (−0.488 + 0.671i)27-s + (−9.31 − 3.02i)29-s + (1.81 + 2.49i)31-s + ⋯
L(s)  = 1  + (1.30 − 0.424i)3-s + (0.00943 + 0.00685i)5-s + (0.0530 − 0.163i)7-s + (0.714 − 0.519i)9-s + (0.940 + 0.340i)11-s + (−0.354 − 0.488i)13-s + (0.0152 + 0.00494i)15-s + (0.470 − 0.647i)17-s + (0.184 + 0.566i)19-s − 0.235i·21-s + 0.770i·23-s + (−0.308 − 0.950i)25-s + (−0.0939 + 0.129i)27-s + (−1.73 − 0.562i)29-s + (0.325 + 0.448i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.906 + 0.421i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.906 + 0.421i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.906 + 0.421i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97335 - 0.435873i\)
\(L(\frac12)\) \(\approx\) \(1.97335 - 0.435873i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.11 - 1.12i)T \)
good3 \( 1 + (-2.26 + 0.734i)T + (2.42 - 1.76i)T^{2} \)
5 \( 1 + (-0.0210 - 0.0153i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.140 + 0.432i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.27 + 1.76i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.94 + 2.67i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.802 - 2.47i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.69iT - 23T^{2} \)
29 \( 1 + (9.31 + 3.02i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.81 - 2.49i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.27 - 7.00i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (3.59 - 1.16i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + 10.2T + 43T^{2} \)
47 \( 1 + (-10.7 + 3.47i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-7.06 + 5.13i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (6.16 + 2.00i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (7.78 - 10.7i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 8.31iT - 67T^{2} \)
71 \( 1 + (-2.37 + 3.26i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.48 + 1.13i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.29 + 1.66i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.28 - 0.934i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + (-0.120 + 0.0875i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.65794466257011388395348569629, −10.19108165941962337241855261041, −9.481712322285158818063682031139, −8.559862231524703747543427585816, −7.69156684892840039003614201630, −6.97514734082246030322455857381, −5.55788881836140918897842394385, −4.04090732274860988021536861619, −3.00460832261600005842325277960, −1.66230075470952485835856993064, 1.96209150963325959685292842980, 3.34503207207568131748700701441, 4.16267140549810298121270510965, 5.62538884593340010985027652882, 6.96396957888349761723800930162, 7.954954574357537728764657013894, 9.069023838473556106727934820584, 9.251884669551878705970602722293, 10.48119967795074845648770327790, 11.51989012916702720551200537656

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