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Properties

Label 2-368-16.13-c1-0-10
Degree $2$
Conductor $368$
Sign $0.515 - 0.856i$
Analytic cond. $2.93849$
Root an. cond. $1.71420$
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  + (−0.741 − 1.20i)2-s + (1.96 + 1.96i)3-s + (−0.900 + 1.78i)4-s + (−0.463 + 0.463i)5-s + (0.908 − 3.81i)6-s + 1.33i·7-s + (2.81 − 0.238i)8-s + 4.69i·9-s + (0.900 + 0.214i)10-s + (−1.06 + 1.06i)11-s + (−5.26 + 1.73i)12-s + (−0.743 − 0.743i)13-s + (1.61 − 0.992i)14-s − 1.81·15-s + (−2.37 − 3.21i)16-s − 1.89·17-s + ⋯
L(s)  = 1  + (−0.524 − 0.851i)2-s + (1.13 + 1.13i)3-s + (−0.450 + 0.892i)4-s + (−0.207 + 0.207i)5-s + (0.370 − 1.55i)6-s + 0.506i·7-s + (0.996 − 0.0843i)8-s + 1.56i·9-s + (0.284 + 0.0678i)10-s + (−0.319 + 0.319i)11-s + (−1.52 + 0.500i)12-s + (−0.206 − 0.206i)13-s + (0.431 − 0.265i)14-s − 0.468·15-s + (−0.594 − 0.804i)16-s − 0.460·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(368\)    =    \(2^{4} \cdot 23\)
Sign: $0.515 - 0.856i$
Analytic conductor: \(2.93849\)
Root analytic conductor: \(1.71420\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{368} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 368,\ (\ :1/2),\ 0.515 - 0.856i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.13585 + 0.641968i\)
\(L(\frac12)\) \(\approx\) \(1.13585 + 0.641968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.741 + 1.20i)T \)
23 \( 1 + iT \)
good3 \( 1 + (-1.96 - 1.96i)T + 3iT^{2} \)
5 \( 1 + (0.463 - 0.463i)T - 5iT^{2} \)
7 \( 1 - 1.33iT - 7T^{2} \)
11 \( 1 + (1.06 - 1.06i)T - 11iT^{2} \)
13 \( 1 + (0.743 + 0.743i)T + 13iT^{2} \)
17 \( 1 + 1.89T + 17T^{2} \)
19 \( 1 + (-4.25 - 4.25i)T + 19iT^{2} \)
29 \( 1 + (2.48 + 2.48i)T + 29iT^{2} \)
31 \( 1 - 1.53T + 31T^{2} \)
37 \( 1 + (0.463 - 0.463i)T - 37iT^{2} \)
41 \( 1 - 1.58iT - 41T^{2} \)
43 \( 1 + (-2.95 + 2.95i)T - 43iT^{2} \)
47 \( 1 - 6.03T + 47T^{2} \)
53 \( 1 + (-7.20 + 7.20i)T - 53iT^{2} \)
59 \( 1 + (-9.14 + 9.14i)T - 59iT^{2} \)
61 \( 1 + (2.74 + 2.74i)T + 61iT^{2} \)
67 \( 1 + (0.407 + 0.407i)T + 67iT^{2} \)
71 \( 1 + 15.9iT - 71T^{2} \)
73 \( 1 - 6.53iT - 73T^{2} \)
79 \( 1 + 9.08T + 79T^{2} \)
83 \( 1 + (11.9 + 11.9i)T + 83iT^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 + 8.92T + 97T^{2} \)
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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

−11.33853755043458139779288064268, −10.36417996121122371142390291609, −9.745528094495017791755197166108, −9.015134756316643859837106046614, −8.201972016641441404492656845976, −7.36315320839416709862940117806, −5.29856743403655241437701460150, −4.10684804262063368598302028222, −3.24128829899481874042825773936, −2.22013890400501111066808199842, 0.989376765244284816075103307449, 2.59007737404211777848627304280, 4.26612886011434553147276122668, 5.72401332789039477256264373987, 7.12902295211146191500378819885, 7.29443617810433892491884802556, 8.437961348230923651974086311381, 8.939670132642160019238627108249, 10.00090594657659349151496040083, 11.17878466305453905424833225076

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