↑

Properties

Label 2-238-119.20-c1-0-9
Degree $2$
Conductor $238$
Sign $0.644 + 0.764i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.382i)2-s + (0.0370 − 0.186i)3-s + (0.707 − 0.707i)4-s + (0.968 − 0.647i)5-s + (−0.0370 − 0.186i)6-s + (−1.59 − 2.11i)7-s + (0.382 − 0.923i)8-s + (2.73 + 1.13i)9-s + (0.647 − 0.968i)10-s + (−0.258 − 1.29i)11-s + (−0.105 − 0.157i)12-s + (−2.00 + 2.00i)13-s + (−2.28 − 1.34i)14-s + (−0.0846 − 0.204i)15-s − i·16-s + (4.08 + 0.541i)17-s + ⋯
L(s)  = 1  + (0.653 − 0.270i)2-s + (0.0213 − 0.107i)3-s + (0.353 − 0.353i)4-s + (0.433 − 0.289i)5-s + (−0.0151 − 0.0760i)6-s + (−0.602 − 0.798i)7-s + (0.135 − 0.326i)8-s + (0.912 + 0.378i)9-s + (0.204 − 0.306i)10-s + (−0.0778 − 0.391i)11-s + (−0.0304 − 0.0455i)12-s + (−0.555 + 0.555i)13-s + (−0.609 − 0.358i)14-s + (−0.0218 − 0.0527i)15-s − 0.250i·16-s + (0.991 + 0.131i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(238\)    =    \(2 \cdot 7 \cdot 17\)
Sign: $0.644 + 0.764i$
Analytic conductor: \(1.90043\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{238} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 238,\ (\ :1/2),\ 0.644 + 0.764i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66519 - 0.773635i\)
\(L(\frac12)\) \(\approx\) \(1.66519 - 0.773635i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (1.59 + 2.11i)T \)
17 \( 1 + (-4.08 - 0.541i)T \)
good3 \( 1 + (-0.0370 + 0.186i)T + (-2.77 - 1.14i)T^{2} \)
5 \( 1 + (-0.968 + 0.647i)T + (1.91 - 4.61i)T^{2} \)
11 \( 1 + (0.258 + 1.29i)T + (-10.1 + 4.20i)T^{2} \)
13 \( 1 + (2.00 - 2.00i)T - 13iT^{2} \)
19 \( 1 + (-0.316 - 0.764i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (3.00 - 0.598i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (4.81 - 3.21i)T + (11.0 - 26.7i)T^{2} \)
31 \( 1 + (-3.15 - 0.628i)T + (28.6 + 11.8i)T^{2} \)
37 \( 1 + (0.927 + 0.184i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (0.173 + 0.115i)T + (15.6 + 37.8i)T^{2} \)
43 \( 1 + (0.387 + 0.160i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 + (6.54 - 6.54i)T - 47iT^{2} \)
53 \( 1 + (5.89 - 2.44i)T + (37.4 - 37.4i)T^{2} \)
59 \( 1 + (-9.52 - 3.94i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (1.53 - 2.29i)T + (-23.3 - 56.3i)T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 + (-1.17 - 0.233i)T + (65.5 + 27.1i)T^{2} \)
73 \( 1 + (-13.1 + 8.77i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (0.712 + 3.58i)T + (-72.9 + 30.2i)T^{2} \)
83 \( 1 + (11.9 - 4.95i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (8.24 + 8.24i)T + 89iT^{2} \)
97 \( 1 + (6.75 + 10.1i)T + (-37.1 + 89.6i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.28155022530922628248693019239, −11.07899864381383954681609645802, −10.07336759127191195600707959475, −9.508292065929003425401555508566, −7.80742733188384467794361378363, −6.90812776887846122396674554875, −5.74167790976901718706975220449, −4.54696750760331643975083099579, −3.39144079611385118189422596811, −1.60058273971430381607644155266, 2.36152794243216483300386475212, 3.68802286432273861516505126328, 5.10850418426119743730155424403, 6.10736519239813950063482369100, 7.05661619141003367357547795337, 8.185885092244001639203711953040, 9.745274148493729323918287850098, 10.03243544728659528052850073066, 11.63565865081566289298962172433, 12.47782914981026549506922166907

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