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Properties

Label 2-945-35.13-c1-0-61
Degree $2$
Conductor $945$
Sign $-0.954 - 0.296i$
Analytic cond. $7.54586$
Root an. cond. $2.74697$
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  + (1.50 − 1.50i)2-s − 2.52i·4-s + (−0.469 − 2.18i)5-s + (−1.94 − 1.79i)7-s + (−0.788 − 0.788i)8-s + (−3.99 − 2.58i)10-s − 4.53·11-s + (−1.52 + 1.52i)13-s + (−5.62 + 0.228i)14-s + 2.67·16-s + (−4.66 − 4.66i)17-s + 2.37·19-s + (−5.51 + 1.18i)20-s + (−6.82 + 6.82i)22-s + (5.47 + 5.47i)23-s + ⋯
L(s)  = 1  + (1.06 − 1.06i)2-s − 1.26i·4-s + (−0.209 − 0.977i)5-s + (−0.735 − 0.677i)7-s + (−0.278 − 0.278i)8-s + (−1.26 − 0.816i)10-s − 1.36·11-s + (−0.423 + 0.423i)13-s + (−1.50 + 0.0610i)14-s + 0.669·16-s + (−1.13 − 1.13i)17-s + 0.544·19-s + (−1.23 + 0.265i)20-s + (−1.45 + 1.45i)22-s + (1.14 + 1.14i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $-0.954 - 0.296i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ -0.954 - 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.261259 + 1.72018i\)
\(L(\frac12)\) \(\approx\) \(0.261259 + 1.72018i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.469 + 2.18i)T \)
7 \( 1 + (1.94 + 1.79i)T \)
good2 \( 1 + (-1.50 + 1.50i)T - 2iT^{2} \)
11 \( 1 + 4.53T + 11T^{2} \)
13 \( 1 + (1.52 - 1.52i)T - 13iT^{2} \)
17 \( 1 + (4.66 + 4.66i)T + 17iT^{2} \)
19 \( 1 - 2.37T + 19T^{2} \)
23 \( 1 + (-5.47 - 5.47i)T + 23iT^{2} \)
29 \( 1 + 7.99iT - 29T^{2} \)
31 \( 1 - 3.44iT - 31T^{2} \)
37 \( 1 + (-5.56 + 5.56i)T - 37iT^{2} \)
41 \( 1 + 6.70iT - 41T^{2} \)
43 \( 1 + (3.21 + 3.21i)T + 43iT^{2} \)
47 \( 1 + (3.76 + 3.76i)T + 47iT^{2} \)
53 \( 1 + (4.25 + 4.25i)T + 53iT^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 - 6.27iT - 61T^{2} \)
67 \( 1 + (2.67 - 2.67i)T - 67iT^{2} \)
71 \( 1 + 1.93T + 71T^{2} \)
73 \( 1 + (3.55 - 3.55i)T - 73iT^{2} \)
79 \( 1 + 15.0iT - 79T^{2} \)
83 \( 1 + (-9.08 + 9.08i)T - 83iT^{2} \)
89 \( 1 - 1.70T + 89T^{2} \)
97 \( 1 + (-3.42 - 3.42i)T + 97iT^{2} \)
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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

−9.782499254108808955162169383377, −9.070311957340574731102934240011, −7.79889311103808299636440134779, −7.06837284718688277334056290976, −5.58839268035632024032840937499, −4.94903468254586801458649179260, −4.18969234568787946418646598308, −3.17306794998019795953594872798, −2.17166826027668604107928375870, −0.54233585804770550501177840912, 2.63737565786887369346418152201, 3.26720865724287721072173611621, 4.57158155656487051194582474284, 5.38097349585007937880411541726, 6.33646689796454821890662859489, 6.76749232374142994478219280918, 7.74802901802166037094554761146, 8.441404126974830497261965439468, 9.761777464900151904909625614456, 10.54080319966598539026147230613

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