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Properties

Label 2-465-15.2-c1-0-20
Degree $2$
Conductor $465$
Sign $-0.999 - 0.0165i$
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  + (−1.74 + 1.74i)2-s + (1.27 + 1.17i)3-s − 4.05i·4-s + (0.384 + 2.20i)5-s + (−4.25 + 0.181i)6-s + (1.30 + 1.30i)7-s + (3.58 + 3.58i)8-s + (0.254 + 2.98i)9-s + (−4.50 − 3.16i)10-s + 2.55i·11-s + (4.75 − 5.17i)12-s + (2.37 − 2.37i)13-s − 4.53·14-s + (−2.09 + 3.26i)15-s − 4.34·16-s + (3.38 − 3.38i)17-s + ⋯
L(s)  = 1  + (−1.23 + 1.23i)2-s + (0.736 + 0.676i)3-s − 2.02i·4-s + (0.171 + 0.985i)5-s + (−1.73 + 0.0739i)6-s + (0.492 + 0.492i)7-s + (1.26 + 1.26i)8-s + (0.0848 + 0.996i)9-s + (−1.42 − 1.00i)10-s + 0.771i·11-s + (1.37 − 1.49i)12-s + (0.657 − 0.657i)13-s − 1.21·14-s + (−0.539 + 0.841i)15-s − 1.08·16-s + (0.821 − 0.821i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.999 - 0.0165i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.999 - 0.0165i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00845111 + 1.02335i\)
\(L(\frac12)\) \(\approx\) \(0.00845111 + 1.02335i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.27 - 1.17i)T \)
5 \( 1 + (-0.384 - 2.20i)T \)
31 \( 1 - T \)
good2 \( 1 + (1.74 - 1.74i)T - 2iT^{2} \)
7 \( 1 + (-1.30 - 1.30i)T + 7iT^{2} \)
11 \( 1 - 2.55iT - 11T^{2} \)
13 \( 1 + (-2.37 + 2.37i)T - 13iT^{2} \)
17 \( 1 + (-3.38 + 3.38i)T - 17iT^{2} \)
19 \( 1 - 4.69iT - 19T^{2} \)
23 \( 1 + (0.268 + 0.268i)T + 23iT^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
37 \( 1 + (2.80 + 2.80i)T + 37iT^{2} \)
41 \( 1 + 3.76iT - 41T^{2} \)
43 \( 1 + (-7.57 + 7.57i)T - 43iT^{2} \)
47 \( 1 + (-2.85 + 2.85i)T - 47iT^{2} \)
53 \( 1 + (-3.33 - 3.33i)T + 53iT^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 + 7.79T + 61T^{2} \)
67 \( 1 + (4.02 + 4.02i)T + 67iT^{2} \)
71 \( 1 + 1.18iT - 71T^{2} \)
73 \( 1 + (-3.86 + 3.86i)T - 73iT^{2} \)
79 \( 1 - 4.46iT - 79T^{2} \)
83 \( 1 + (-4.02 - 4.02i)T + 83iT^{2} \)
89 \( 1 - 16.4T + 89T^{2} \)
97 \( 1 + (-13.6 - 13.6i)T + 97iT^{2} \)
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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

−10.79293556783297163009871593971, −10.31529952645934507975395267411, −9.488632234408920506643627457202, −8.790551808657862374708270494542, −7.68927142556707211853726832676, −7.44557172471869059494160421339, −6.01454740864737710936861402971, −5.25204365723950755451193581447, −3.55690415239960541164334848931, −2.01038281850584836998431747017, 0.953535804042625335778095538269, 1.74805931138019680239826988959, 3.17244216466687800103715057560, 4.25736447770304102333788173838, 6.07654081504705076853312995905, 7.58577323081133389990893346992, 8.155175736409027237882057296383, 8.983290645161435957964044380318, 9.393209997664870257825760290382, 10.60590143055337380021109799671

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