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Properties

Label 2-675-45.38-c1-0-11
Degree $2$
Conductor $675$
Sign $0.759 + 0.649i$
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  + (0.186 − 0.0499i)2-s + (−1.69 + 0.981i)4-s + (−0.632 − 2.35i)7-s + (−0.540 + 0.540i)8-s + (2.14 + 1.23i)11-s + (0.422 − 1.57i)13-s + (−0.235 − 0.407i)14-s + (1.88 − 3.27i)16-s + (0.403 + 0.403i)17-s − 4.28i·19-s + (0.461 + 0.123i)22-s + (6.82 + 1.82i)23-s − 0.314i·26-s + (3.38 + 3.38i)28-s + (3.20 − 5.55i)29-s + ⋯
L(s)  = 1  + (0.131 − 0.0352i)2-s + (−0.849 + 0.490i)4-s + (−0.238 − 0.891i)7-s + (−0.191 + 0.191i)8-s + (0.646 + 0.373i)11-s + (0.117 − 0.436i)13-s + (−0.0629 − 0.108i)14-s + (0.472 − 0.818i)16-s + (0.0979 + 0.0979i)17-s − 0.983i·19-s + (0.0982 + 0.0263i)22-s + (1.42 + 0.381i)23-s − 0.0616i·26-s + (0.640 + 0.640i)28-s + (0.595 − 1.03i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.759 + 0.649i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.759 + 0.649i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17143 - 0.432629i\)
\(L(\frac12)\) \(\approx\) \(1.17143 - 0.432629i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.186 + 0.0499i)T + (1.73 - i)T^{2} \)
7 \( 1 + (0.632 + 2.35i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-2.14 - 1.23i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.422 + 1.57i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-0.403 - 0.403i)T + 17iT^{2} \)
19 \( 1 + 4.28iT - 19T^{2} \)
23 \( 1 + (-6.82 - 1.82i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (-3.20 + 5.55i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.97 + 3.41i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.171 + 0.171i)T - 37iT^{2} \)
41 \( 1 + (-6.52 + 3.76i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.95 + 1.32i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (-2.91 + 0.780i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.12 - 6.12i)T - 53iT^{2} \)
59 \( 1 + (-2.27 - 3.93i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.235 - 0.408i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.65 + 0.443i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.50iT - 71T^{2} \)
73 \( 1 + (6.88 + 6.88i)T + 73iT^{2} \)
79 \( 1 + (6.50 + 3.75i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.85 + 10.6i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 2.90T + 89T^{2} \)
97 \( 1 + (-0.379 - 1.41i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.35307282170624129590502853293, −9.400573079260738621393272219397, −8.852586002006849153354328225642, −7.67279270246272994621138018794, −7.08216709310642488721114850939, −5.82526689840498368490518882027, −4.63948948688814260436254894230, −3.96323446101768105959895325389, −2.85754787984749779763543447135, −0.78239428798224564757232387171, 1.29674075438433940760597739475, 3.01665539073572654136175458683, 4.16132567138348891354045254541, 5.20645097879393812733239424065, 5.98861079016115959753929577149, 6.87060830308952417432935422015, 8.292821991652751940808333077851, 8.989939822392393404436504547178, 9.509657120341251626700404226735, 10.55055039062749046287673187510

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