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Properties

Label 2-675-9.4-c1-0-10
Degree $2$
Conductor $675$
Sign $0.333 + 0.942i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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  + (−0.285 − 0.495i)2-s + (0.836 − 1.44i)4-s + (0.714 + 1.23i)7-s − 2.10·8-s + (1.33 + 2.31i)11-s + (2.33 − 4.04i)13-s + (0.408 − 0.707i)14-s + (−1.07 − 1.85i)16-s + 2.67·17-s + 4.67·19-s + (0.764 − 1.32i)22-s + (2.95 − 5.12i)23-s − 2.67·26-s + 2.38·28-s + (−4.74 − 8.21i)29-s + ⋯
L(s)  = 1  + (−0.202 − 0.350i)2-s + (0.418 − 0.724i)4-s + (0.269 + 0.467i)7-s − 0.742·8-s + (0.402 + 0.697i)11-s + (0.648 − 1.12i)13-s + (0.109 − 0.189i)14-s + (−0.267 − 0.464i)16-s + 0.648·17-s + 1.07·19-s + (0.162 − 0.282i)22-s + (0.616 − 1.06i)23-s − 0.524·26-s + 0.451·28-s + (−0.881 − 1.52i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.333 + 0.942i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.333 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28538 - 0.908352i\)
\(L(\frac12)\) \(\approx\) \(1.28538 - 0.908352i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (0.285 + 0.495i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.714 - 1.23i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.33 - 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.33 + 4.04i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.67T + 17T^{2} \)
19 \( 1 - 4.67T + 19T^{2} \)
23 \( 1 + (-2.95 + 5.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.74 + 8.21i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.48 - 6.02i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.81T + 37T^{2} \)
41 \( 1 + (0.735 - 1.27i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.235 - 0.408i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.47 + 6.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.14T + 53T^{2} \)
59 \( 1 + (0.571 - 0.990i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.26 - 2.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.29 - 5.70i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.8T + 71T^{2} \)
73 \( 1 - 1.71T + 73T^{2} \)
79 \( 1 + (-0.143 - 0.249i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.14 - 3.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (-3.91 - 6.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.29368997413160756989184078883, −9.652110827173194775699760142826, −8.758740345126331376872897295566, −7.74822216342740052450693086934, −6.73009801683293069030038149715, −5.73590099541604399322310653752, −5.06038852451021414638018842775, −3.49258447037491906021984847057, −2.32630054808145339630130504395, −1.03262825331258798862399361285, 1.49364915273591953502380306065, 3.22122422569522911619692401394, 3.91279426596856913573648175749, 5.39227794385416358831091265240, 6.39133630326176267375337801986, 7.29607535368830955247128006003, 7.86922721177539388994256228431, 8.988471055499903159457907745028, 9.465215213945976515276710488163, 11.04037013250849574621302076097

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