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Properties

Label 2-352-44.7-c1-0-8
Degree $2$
Conductor $352$
Sign $0.998 - 0.0505i$
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  + (2.56 + 0.832i)3-s + (1.18 − 0.862i)5-s + (−0.845 − 2.60i)7-s + (3.45 + 2.50i)9-s + (−0.122 + 3.31i)11-s + (1.50 − 2.06i)13-s + (3.76 − 1.22i)15-s + (−3.48 − 4.79i)17-s + (−0.531 + 1.63i)19-s − 7.37i·21-s + 7.75i·23-s + (−0.880 + 2.70i)25-s + (2.00 + 2.76i)27-s + (−3.07 + 0.997i)29-s + (−3.93 + 5.42i)31-s + ⋯
L(s)  = 1  + (1.48 + 0.480i)3-s + (0.530 − 0.385i)5-s + (−0.319 − 0.983i)7-s + (1.15 + 0.835i)9-s + (−0.0369 + 0.999i)11-s + (0.417 − 0.574i)13-s + (0.971 − 0.315i)15-s + (−0.844 − 1.16i)17-s + (−0.121 + 0.375i)19-s − 1.60i·21-s + 1.61i·23-s + (−0.176 + 0.541i)25-s + (0.385 + 0.531i)27-s + (−0.570 + 0.185i)29-s + (−0.707 + 0.973i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0505i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(352\)    =    \(2^{5} \cdot 11\)
Sign: $0.998 - 0.0505i$
Analytic conductor: \(2.81073\)
Root analytic conductor: \(1.67652\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{352} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 352,\ (\ :1/2),\ 0.998 - 0.0505i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16548 + 0.0547556i\)
\(L(\frac12)\) \(\approx\) \(2.16548 + 0.0547556i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.122 - 3.31i)T \)
good3 \( 1 + (-2.56 - 0.832i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.18 + 0.862i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.845 + 2.60i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.50 + 2.06i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.48 + 4.79i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.531 - 1.63i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 7.75iT - 23T^{2} \)
29 \( 1 + (3.07 - 0.997i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.93 - 5.42i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.50 + 7.70i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-10.8 - 3.52i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 - 0.0675T + 43T^{2} \)
47 \( 1 + (6.48 + 2.10i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.99 + 3.62i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-5.13 + 1.66i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-4.35 - 5.99i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 11.2iT - 67T^{2} \)
71 \( 1 + (-5.04 - 6.94i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (15.1 - 4.92i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (8.56 + 6.22i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.450 - 0.327i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 13.1T + 89T^{2} \)
97 \( 1 + (0.746 + 0.542i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.29123055062883657630059725245, −10.20842785846865768830754598967, −9.496450594791890679970039515781, −8.985354631554722030447876899555, −7.70001059263530283707714516799, −7.10573480434801295877987939103, −5.39373115561712873101683586035, −4.17735498953195049301956141062, −3.25836737208530271451225917859, −1.83002677554276318407433452184, 2.05926427926641434155925742212, 2.80904908156968899702677245021, 4.07631121588351568947015203578, 5.98249428484381266271284332054, 6.59109518452063345901097151783, 8.006784860305837168984941955116, 8.749678129023686787125788548062, 9.212708452296337732544739952483, 10.44616462391110388524477588898, 11.46298386423063321190381501479

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