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Properties

Label 2-238-119.53-c1-0-6
Degree $2$
Conductor $238$
Sign $0.557 + 0.830i$
Analytic cond. $1.90043$
Root an. cond. $1.37856$
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  + (0.258 − 0.965i)2-s + (−0.0571 + 0.433i)3-s + (−0.866 − 0.499i)4-s + (−0.465 + 0.607i)5-s + (0.404 + 0.167i)6-s + (2.01 − 1.72i)7-s + (−0.707 + 0.707i)8-s + (2.71 + 0.726i)9-s + (0.465 + 0.607i)10-s + (3.31 − 2.54i)11-s + (0.266 − 0.347i)12-s − 4.16i·13-s + (−1.14 − 2.38i)14-s + (−0.236 − 0.236i)15-s + (0.500 + 0.866i)16-s + (−3.97 − 1.09i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.0329 + 0.250i)3-s + (−0.433 − 0.249i)4-s + (−0.208 + 0.271i)5-s + (0.165 + 0.0683i)6-s + (0.759 − 0.650i)7-s + (−0.249 + 0.249i)8-s + (0.904 + 0.242i)9-s + (0.147 + 0.192i)10-s + (0.999 − 0.766i)11-s + (0.0768 − 0.100i)12-s − 1.15i·13-s + (−0.305 − 0.637i)14-s + (−0.0611 − 0.0611i)15-s + (0.125 + 0.216i)16-s + (−0.964 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.557 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(238\)    =    \(2 \cdot 7 \cdot 17\)
Sign: $0.557 + 0.830i$
Analytic conductor: \(1.90043\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{238} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 238,\ (\ :1/2),\ 0.557 + 0.830i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23781 - 0.659542i\)
\(L(\frac12)\) \(\approx\) \(1.23781 - 0.659542i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 + (-2.01 + 1.72i)T \)
17 \( 1 + (3.97 + 1.09i)T \)
good3 \( 1 + (0.0571 - 0.433i)T + (-2.89 - 0.776i)T^{2} \)
5 \( 1 + (0.465 - 0.607i)T + (-1.29 - 4.82i)T^{2} \)
11 \( 1 + (-3.31 + 2.54i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + 4.16iT - 13T^{2} \)
19 \( 1 + (0.301 - 1.12i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.19 - 9.10i)T + (-22.2 + 5.95i)T^{2} \)
29 \( 1 + (1.45 + 3.50i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-0.557 + 4.23i)T + (-29.9 - 8.02i)T^{2} \)
37 \( 1 + (2.57 + 1.97i)T + (9.57 + 35.7i)T^{2} \)
41 \( 1 + (0.725 - 1.75i)T + (-28.9 - 28.9i)T^{2} \)
43 \( 1 + (5.38 - 5.38i)T - 43iT^{2} \)
47 \( 1 + (10.1 - 5.84i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.63 + 2.31i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-2.39 - 8.95i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-5.28 + 0.696i)T + (58.9 - 15.7i)T^{2} \)
67 \( 1 + (-2.25 + 3.90i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (12.9 - 5.35i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (14.0 + 1.84i)T + (70.5 + 18.8i)T^{2} \)
79 \( 1 + (0.881 + 6.69i)T + (-76.3 + 20.4i)T^{2} \)
83 \( 1 + (-11.2 - 11.2i)T + 83iT^{2} \)
89 \( 1 + (12.4 - 7.18i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.73 + 6.60i)T + (-68.5 + 68.5i)T^{2} \)
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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

−11.59323477951844640639106031759, −11.21890168487922288457884602539, −10.25169081833887897117806920615, −9.340607297225848609596445179984, −8.075994769655976383104853888196, −7.08906329306670559904621296011, −5.55722361571606608693115294275, −4.33795229648597732396179309893, −3.40095753911400446004927168375, −1.41721544974142244765497936088, 1.87110188483661652297845966488, 4.22112031038137298865258131400, 4.81887285386500631067606961562, 6.59319820598718011112977958041, 6.94677311218706207297160222871, 8.520344276023389954741929233812, 8.963234224227431426633062337768, 10.28587406609964163677932582646, 11.71626942621318478406292276091, 12.28007403035995390737641607835

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