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Properties

Label 2-945-315.202-c1-0-22
Degree $2$
Conductor $945$
Sign $0.919 + 0.393i$
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  + (0.377 − 1.41i)2-s + (−0.113 − 0.0657i)4-s + (2.18 + 0.477i)5-s + (−0.573 + 2.58i)7-s + (1.92 − 1.92i)8-s + (1.49 − 2.89i)10-s + (1.95 + 3.38i)11-s + (0.354 − 0.0948i)13-s + (3.42 + 1.78i)14-s + (−2.12 − 3.67i)16-s + (−1.41 + 1.41i)17-s − 3.83·19-s + (−0.217 − 0.198i)20-s + (5.51 − 1.47i)22-s + (1.67 + 6.23i)23-s + ⋯
L(s)  = 1  + (0.267 − 0.997i)2-s + (−0.0569 − 0.0328i)4-s + (0.976 + 0.213i)5-s + (−0.216 + 0.976i)7-s + (0.681 − 0.681i)8-s + (0.474 − 0.917i)10-s + (0.589 + 1.02i)11-s + (0.0981 − 0.0263i)13-s + (0.915 + 0.477i)14-s + (−0.530 − 0.919i)16-s + (−0.342 + 0.342i)17-s − 0.879·19-s + (−0.0486 − 0.0443i)20-s + (1.17 − 0.314i)22-s + (0.348 + 1.30i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(945\)    =    \(3^{3} \cdot 5 \cdot 7\)
Sign: $0.919 + 0.393i$
Analytic conductor: \(7.54586\)
Root analytic conductor: \(2.74697\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{945} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 945,\ (\ :1/2),\ 0.919 + 0.393i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.38419 - 0.489211i\)
\(L(\frac12)\) \(\approx\) \(2.38419 - 0.489211i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.18 - 0.477i)T \)
7 \( 1 + (0.573 - 2.58i)T \)
good2 \( 1 + (-0.377 + 1.41i)T + (-1.73 - i)T^{2} \)
11 \( 1 + (-1.95 - 3.38i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.354 + 0.0948i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.41 - 1.41i)T - 17iT^{2} \)
19 \( 1 + 3.83T + 19T^{2} \)
23 \( 1 + (-1.67 - 6.23i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.97 + 2.29i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.80 + 1.04i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.0303 + 0.0303i)T + 37iT^{2} \)
41 \( 1 + (9.89 + 5.71i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.456 - 0.122i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (-11.3 - 3.03i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-8.71 + 8.71i)T - 53iT^{2} \)
59 \( 1 + (-4.90 + 8.49i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.75 - 3.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.81 - 1.02i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 2.12T + 71T^{2} \)
73 \( 1 + (8.94 + 8.94i)T + 73iT^{2} \)
79 \( 1 + (3.15 - 1.82i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.53 + 13.1i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.00T + 89T^{2} \)
97 \( 1 + (-11.7 - 3.15i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.10966032074545935352184247859, −9.401039136168980274136361345861, −8.672345395193958516958151179013, −7.26202126920958888389670816912, −6.53706495063042369664360139658, −5.60433778874242541537365876099, −4.51264134912841220584385885986, −3.39772123809853241738301586761, −2.29361001936179255071848210884, −1.70920790584971998020243084327, 1.16216957228634670817778721329, 2.62779063817075995972217566876, 4.13195617847806297003719635827, 5.01299904255911263173960665063, 6.05052671221605084958222478941, 6.56169052648265635406296823151, 7.24045246447472845804158661152, 8.504858027957854253909145502356, 8.931899685629427270042182888413, 10.37632768440936228229690786029

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