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Properties

Label 2-238-119.97-c1-0-5
Degree $2$
Conductor $238$
Sign $-0.217 - 0.976i$
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 + (1.67 + 2.51i)3-s + (−0.707 + 0.707i)4-s + (0.693 − 3.48i)5-s + (−1.67 + 2.51i)6-s + (−0.600 + 2.57i)7-s + (−0.923 − 0.382i)8-s + (−2.35 + 5.67i)9-s + (3.48 − 0.693i)10-s + (−1.40 + 2.09i)11-s + (−2.96 − 0.589i)12-s + (3.64 − 3.64i)13-s + (−2.61 + 0.431i)14-s + (9.92 − 4.11i)15-s − i·16-s + (−4.05 − 0.757i)17-s + ⋯
L(s)  = 1  + (0.270 + 0.653i)2-s + (0.969 + 1.45i)3-s + (−0.353 + 0.353i)4-s + (0.310 − 1.55i)5-s + (−0.685 + 1.02i)6-s + (−0.226 + 0.973i)7-s + (−0.326 − 0.135i)8-s + (−0.783 + 1.89i)9-s + (1.10 − 0.219i)10-s + (−0.422 + 0.632i)11-s + (−0.856 − 0.170i)12-s + (1.01 − 1.01i)13-s + (−0.697 + 0.115i)14-s + (2.56 − 1.06i)15-s − 0.250i·16-s + (−0.982 − 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(238\)    =    \(2 \cdot 7 \cdot 17\)
Sign: $-0.217 - 0.976i$
Analytic conductor: \(1.90043\)
Root analytic conductor: \(1.37856\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{238} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 238,\ (\ :1/2),\ -0.217 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12947 + 1.40910i\)
\(L(\frac12)\) \(\approx\) \(1.12947 + 1.40910i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.600 - 2.57i)T \)
17 \( 1 + (4.05 + 0.757i)T \)
good3 \( 1 + (-1.67 - 2.51i)T + (-1.14 + 2.77i)T^{2} \)
5 \( 1 + (-0.693 + 3.48i)T + (-4.61 - 1.91i)T^{2} \)
11 \( 1 + (1.40 - 2.09i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 + (-3.64 + 3.64i)T - 13iT^{2} \)
19 \( 1 + (-4.92 + 2.04i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (2.62 + 1.75i)T + (8.80 + 21.2i)T^{2} \)
29 \( 1 + (-0.944 + 4.74i)T + (-26.7 - 11.0i)T^{2} \)
31 \( 1 + (-1.06 + 0.712i)T + (11.8 - 28.6i)T^{2} \)
37 \( 1 + (-6.18 + 4.13i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (-0.160 - 0.805i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (1.47 - 3.55i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + (4.53 - 4.53i)T - 47iT^{2} \)
53 \( 1 + (-0.797 - 1.92i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (-3.30 + 7.98i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-0.586 + 0.116i)T + (56.3 - 23.3i)T^{2} \)
67 \( 1 - 4.80iT - 67T^{2} \)
71 \( 1 + (11.0 - 7.35i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (2.07 - 10.4i)T + (-67.4 - 27.9i)T^{2} \)
79 \( 1 + (3.28 - 4.91i)T + (-30.2 - 72.9i)T^{2} \)
83 \( 1 + (3.26 + 7.88i)T + (-58.6 + 58.6i)T^{2} \)
89 \( 1 + (1.35 + 1.35i)T + 89iT^{2} \)
97 \( 1 + (-2.13 - 0.424i)T + (89.6 + 37.1i)T^{2} \)
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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

−12.88861377871659793201328711863, −11.49239567450233893845649480538, −9.964766526518455778293575375429, −9.333894913883298412346688617146, −8.605272196999038570867905772350, −7.997288704818920600149030471731, −5.84634745739303617681657683285, −5.02676239449289960898334124458, −4.21546745734545293756758601795, −2.71990646981923133558391332448, 1.60508413753585844494892767168, 2.90739330749891530539844272827, 3.70161505575075531499374029075, 6.19227202262115009796180173984, 6.82721391004165248350102939539, 7.74681134582445126155812199582, 8.909261439007218024517988501251, 10.12554070035965424354703434222, 11.05740963369833344479603837531, 11.82716265792841255988222789543

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