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Properties

Label 2-238-119.41-c1-0-6
Degree $2$
Conductor $238$
Sign $-0.00699 + 0.999i$
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  + (−0.382 + 0.923i)2-s + (−0.227 − 0.151i)3-s + (−0.707 − 0.707i)4-s + (−2.10 + 0.418i)5-s + (0.227 − 0.151i)6-s + (−1.07 − 2.41i)7-s + (0.923 − 0.382i)8-s + (−1.11 − 2.70i)9-s + (0.418 − 2.10i)10-s + (0.290 − 0.194i)11-s + (0.0532 + 0.267i)12-s + (−1.02 − 1.02i)13-s + (2.64 − 0.0704i)14-s + (0.541 + 0.224i)15-s + i·16-s + (3.43 − 2.28i)17-s + ⋯
L(s)  = 1  + (−0.270 + 0.653i)2-s + (−0.131 − 0.0875i)3-s + (−0.353 − 0.353i)4-s + (−0.941 + 0.187i)5-s + (0.0926 − 0.0619i)6-s + (−0.407 − 0.913i)7-s + (0.326 − 0.135i)8-s + (−0.373 − 0.900i)9-s + (0.132 − 0.666i)10-s + (0.0877 − 0.0586i)11-s + (0.0153 + 0.0773i)12-s + (−0.285 − 0.285i)13-s + (0.706 − 0.0188i)14-s + (0.139 + 0.0579i)15-s + 0.250i·16-s + (0.832 − 0.554i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00699 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00699 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(238\)    =    \(2 \cdot 7 \cdot 17\)
Sign: $-0.00699 + 0.999i$
Analytic conductor: \(1.90043\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{238} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 238,\ (\ :1/2),\ -0.00699 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332537 - 0.334871i\)
\(L(\frac12)\) \(\approx\) \(0.332537 - 0.334871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
7 \( 1 + (1.07 + 2.41i)T \)
17 \( 1 + (-3.43 + 2.28i)T \)
good3 \( 1 + (0.227 + 0.151i)T + (1.14 + 2.77i)T^{2} \)
5 \( 1 + (2.10 - 0.418i)T + (4.61 - 1.91i)T^{2} \)
11 \( 1 + (-0.290 + 0.194i)T + (4.20 - 10.1i)T^{2} \)
13 \( 1 + (1.02 + 1.02i)T + 13iT^{2} \)
19 \( 1 + (6.61 + 2.74i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-0.525 - 0.785i)T + (-8.80 + 21.2i)T^{2} \)
29 \( 1 + (0.856 - 0.170i)T + (26.7 - 11.0i)T^{2} \)
31 \( 1 + (-1.57 + 2.35i)T + (-11.8 - 28.6i)T^{2} \)
37 \( 1 + (1.73 - 2.59i)T + (-14.1 - 34.1i)T^{2} \)
41 \( 1 + (-8.32 - 1.65i)T + (37.8 + 15.6i)T^{2} \)
43 \( 1 + (2.47 + 5.98i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + (4.25 + 4.25i)T + 47iT^{2} \)
53 \( 1 + (2.77 - 6.70i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-0.263 - 0.635i)T + (-41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.93 - 9.74i)T + (-56.3 - 23.3i)T^{2} \)
67 \( 1 + 7.81iT - 67T^{2} \)
71 \( 1 + (5.40 - 8.09i)T + (-27.1 - 65.5i)T^{2} \)
73 \( 1 + (-8.54 + 1.69i)T + (67.4 - 27.9i)T^{2} \)
79 \( 1 + (-11.1 + 7.46i)T + (30.2 - 72.9i)T^{2} \)
83 \( 1 + (-6.09 + 14.7i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (0.0673 - 0.0673i)T - 89iT^{2} \)
97 \( 1 + (3.00 + 15.1i)T + (-89.6 + 37.1i)T^{2} \)
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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

−11.87648457431735856580334394538, −10.91008157552272533401901044581, −9.900622820245212954922706504573, −8.859473817188223827833552624698, −7.74587397114767395732411581118, −7.00245504874707871446860002498, −6.02691122670276895359752744687, −4.46557131535159291825002716212, −3.36014210677795055207901443749, −0.41529270521331604594225291356, 2.22860121427435547112852174014, 3.68835176918771220013685340086, 4.90844263743359965086438785029, 6.23421304725162534116487182657, 7.87255668277036137456971638666, 8.419162470658074770215845780615, 9.552688078226156468822281951812, 10.61874812525001935162573826213, 11.43337990104711478960132302949, 12.34915891205960553597817490095

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