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Properties

Label 2-465-93.92-c1-0-17
Degree $2$
Conductor $465$
Sign $0.983 + 0.180i$
Analytic cond. $3.71304$
Root an. cond. $1.92692$
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  − 1.57i·2-s + (0.222 + 1.71i)3-s − 0.472·4-s + i·5-s + (2.70 − 0.349i)6-s + 3.73·7-s − 2.40i·8-s + (−2.90 + 0.764i)9-s + 1.57·10-s − 0.292·11-s + (−0.105 − 0.811i)12-s + 5.20i·13-s − 5.86i·14-s + (−1.71 + 0.222i)15-s − 4.72·16-s + 3.32·17-s + ⋯
L(s)  = 1  − 1.11i·2-s + (0.128 + 0.991i)3-s − 0.236·4-s + 0.447i·5-s + (1.10 − 0.142i)6-s + 1.41·7-s − 0.849i·8-s + (−0.966 + 0.254i)9-s + 0.497·10-s − 0.0881·11-s + (−0.0303 − 0.234i)12-s + 1.44i·13-s − 1.56i·14-s + (−0.443 + 0.0574i)15-s − 1.18·16-s + 0.805·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.983 + 0.180i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.983 + 0.180i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76065 - 0.159937i\)
\(L(\frac12)\) \(\approx\) \(1.76065 - 0.159937i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.222 - 1.71i)T \)
5 \( 1 - iT \)
31 \( 1 + (-5.30 - 1.69i)T \)
good2 \( 1 + 1.57iT - 2T^{2} \)
7 \( 1 - 3.73T + 7T^{2} \)
11 \( 1 + 0.292T + 11T^{2} \)
13 \( 1 - 5.20iT - 13T^{2} \)
17 \( 1 - 3.32T + 17T^{2} \)
19 \( 1 - 1.40T + 19T^{2} \)
23 \( 1 + 3.97T + 23T^{2} \)
29 \( 1 - 7.40T + 29T^{2} \)
37 \( 1 + 8.03iT - 37T^{2} \)
41 \( 1 - 4.58iT - 41T^{2} \)
43 \( 1 + 2.56iT - 43T^{2} \)
47 \( 1 + 4.02iT - 47T^{2} \)
53 \( 1 + 12.0T + 53T^{2} \)
59 \( 1 - 9.26iT - 59T^{2} \)
61 \( 1 + 0.340iT - 61T^{2} \)
67 \( 1 + 10.5T + 67T^{2} \)
71 \( 1 + 12.6iT - 71T^{2} \)
73 \( 1 + 5.57iT - 73T^{2} \)
79 \( 1 - 0.0690iT - 79T^{2} \)
83 \( 1 + 0.951T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 - 6.62T + 97T^{2} \)
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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

−10.99298648248187304344675697713, −10.32862965470486737426530798851, −9.569655769200068607876276289455, −8.581757138179156471638815042854, −7.54252589094058537381050054776, −6.21515296919017084051247859840, −4.80754163723000574603103209279, −4.07626472263390037540688432273, −2.88443351433833360034686445557, −1.71375418585576825680797590197, 1.30418894697673851864961102182, 2.78206594584397310478243066985, 4.82895302655184990439254245696, 5.57589759136770604818455941199, 6.44632223601935523928135861991, 7.75861802582599307158930859160, 7.953839192281299464231446720870, 8.589994602342128112772852991044, 10.14617075601905996802835486141, 11.32179450263008813287652804872

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