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Properties

Label 1-296-296.3-r1-0-0
Degree $1$
Conductor $296$
Sign $-0.313 + 0.949i$
Analytic cond. $31.8096$
Root an. cond. $31.8096$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 − 0.984i)3-s + (0.766 + 0.642i)5-s + (−0.766 − 0.642i)7-s + (−0.939 − 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)15-s + (0.939 + 0.342i)17-s + (−0.173 + 0.984i)19-s + (−0.766 + 0.642i)21-s + (−0.5 + 0.866i)23-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (−0.5 − 0.866i)29-s + 31-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (0.766 + 0.642i)5-s + (−0.766 − 0.642i)7-s + (−0.939 − 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.939 + 0.342i)13-s + (0.766 − 0.642i)15-s + (0.939 + 0.342i)17-s + (−0.173 + 0.984i)19-s + (−0.766 + 0.642i)21-s + (−0.5 + 0.866i)23-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (−0.5 − 0.866i)29-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(296\)    =    \(2^{3} \cdot 37\)
Sign: $-0.313 + 0.949i$
Analytic conductor: \(31.8096\)
Root analytic conductor: \(31.8096\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{296} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 296,\ (1:\ ),\ -0.313 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2361842106 + 0.3266538755i\)
\(L(\frac12)\) \(\approx\) \(0.2361842106 + 0.3266538755i\)
\(L(1)\) \(\approx\) \(0.8622635582 - 0.1791463383i\)
\(L(1)\) \(\approx\) \(0.8622635582 - 0.1791463383i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
37 \( 1 \)
good3 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 + (0.766 + 0.642i)T \)
7 \( 1 + (-0.766 - 0.642i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.939 + 0.342i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 - T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.173 + 0.984i)T \)
73 \( 1 + T \)
79 \( 1 + (0.766 + 0.642i)T \)
83 \( 1 + (-0.939 - 0.342i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.28341040834682412114070655301, −24.146279760702768550636217841331, −22.82581376025193074294801916742, −22.10661326714312656228876196859, −21.38342479259759935536974376451, −20.45376860878287629382017779870, −19.82161484306378305167718413408, −18.53391085680387097291786236441, −17.41239264366998728581619656704, −16.65594566118812887156189597468, −15.74346768554783791423693067593, −14.96487476787794488081730219286, −13.92562455062376467411015456415, −12.76325286748785762063305553816, −12.07798713033859027481367794159, −10.47451626683709035828099615932, −9.76016072113446212968182077091, −9.17212289115034315789843600577, −8.022746321996174622540599658581, −6.487319761726978038575381375587, −5.247413888747130632511835422303, −4.76237289948071591150143493124, −3.10793994967466289606578140487, −2.21645056202552063430495362010, −0.11046175602123808313888553811, 1.42707030194618765397816301273, 2.668564112237943023336376404991, 3.580044212648053666389768177080, 5.58811403637576508281099300045, 6.30526428616368597270274898773, 7.29421416456316480160897339412, 8.151633468991009190932199905439, 9.639860309922853179728429783552, 10.30950648579787698681527970936, 11.60632497665436040160376656771, 12.62363913789220054971525336773, 13.6538732012490442367586319446, 14.014050958160466267535697396395, 15.166288395759436316310802185359, 16.74347675616721933380777316546, 17.16569287479366348309958536098, 18.478226325093323029386131881412, 18.965107519551160802877557669431, 19.7976212200756032381386301282, 20.9996271440192061995624084321, 21.91149500986332156669325266166, 22.95678868781200067083736520600, 23.607321179778893567035901259007, 24.67329293847488825795742784347, 25.44174334083230200816264692205

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