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Properties

Label 2-152-8.5-c1-0-0
Degree $2$
Conductor $152$
Sign $-0.321 - 0.946i$
Analytic cond. $1.21372$
Root an. cond. $1.10169$
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  + (−0.836 − 1.14i)2-s + 3.13i·3-s + (−0.601 + 1.90i)4-s − 0.594i·5-s + (3.57 − 2.62i)6-s − 3.48·7-s + (2.67 − 0.908i)8-s − 6.83·9-s + (−0.677 + 0.496i)10-s + 4.83i·11-s + (−5.98 − 1.88i)12-s + 0.215i·13-s + (2.91 + 3.97i)14-s + 1.86·15-s + (−3.27 − 2.29i)16-s + 1.29·17-s + ⋯
L(s)  = 1  + (−0.591 − 0.806i)2-s + 1.81i·3-s + (−0.300 + 0.953i)4-s − 0.265i·5-s + (1.46 − 1.07i)6-s − 1.31·7-s + (0.946 − 0.321i)8-s − 2.27·9-s + (−0.214 + 0.157i)10-s + 1.45i·11-s + (−1.72 − 0.544i)12-s + 0.0597i·13-s + (0.779 + 1.06i)14-s + 0.481·15-s + (−0.819 − 0.573i)16-s + 0.314·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(152\)    =    \(2^{3} \cdot 19\)
Sign: $-0.321 - 0.946i$
Analytic conductor: \(1.21372\)
Root analytic conductor: \(1.10169\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{152} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 152,\ (\ :1/2),\ -0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349360 + 0.487474i\)
\(L(\frac12)\) \(\approx\) \(0.349360 + 0.487474i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.836 + 1.14i)T \)
19 \( 1 - iT \)
good3 \( 1 - 3.13iT - 3T^{2} \)
5 \( 1 + 0.594iT - 5T^{2} \)
7 \( 1 + 3.48T + 7T^{2} \)
11 \( 1 - 4.83iT - 11T^{2} \)
13 \( 1 - 0.215iT - 13T^{2} \)
17 \( 1 - 1.29T + 17T^{2} \)
23 \( 1 - 4.52T + 23T^{2} \)
29 \( 1 - 9.41iT - 29T^{2} \)
31 \( 1 + 1.22T + 31T^{2} \)
37 \( 1 + 5.62iT - 37T^{2} \)
41 \( 1 + 0.450T + 41T^{2} \)
43 \( 1 + 0.794iT - 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 + 2.56iT - 53T^{2} \)
59 \( 1 + 2.75iT - 59T^{2} \)
61 \( 1 - 7.76iT - 61T^{2} \)
67 \( 1 + 4.11iT - 67T^{2} \)
71 \( 1 + 7.82T + 71T^{2} \)
73 \( 1 - 3.08T + 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 - 11.6iT - 83T^{2} \)
89 \( 1 - 13.7T + 89T^{2} \)
97 \( 1 - 2.08T + 97T^{2} \)
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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

−12.90336569684543100016339315115, −12.20255608223847819878301343460, −10.80925973155525584119392886843, −10.21399535915807316306328720178, −9.365149259213616113809080007751, −8.895676213887488604267746532477, −7.10170875296083044785790003380, −5.14251561103758334636374019504, −4.02449658346951190132811066181, −2.96289491803091372956333676219, 0.71644535900500256772558095458, 2.89171658228689230911127777542, 5.78464516645214554651804505589, 6.42657762674701417407221137817, 7.25205708064039865705391481566, 8.289796641651220211782637726166, 9.202699945305935602128747998538, 10.66058561528742894861105192135, 11.78253807888632868636753658044, 13.10440533774851807962495693044

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