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Properties

Label 2-675-27.22-c1-0-40
Degree $2$
Conductor $675$
Sign $0.554 + 0.831i$
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  + (−0.948 + 0.795i)2-s + (1.62 − 0.595i)3-s + (−0.0810 + 0.459i)4-s + (−1.06 + 1.85i)6-s + (−0.395 − 2.24i)7-s + (−1.52 − 2.64i)8-s + (2.28 − 1.93i)9-s + (−4.87 − 1.77i)11-s + (0.142 + 0.795i)12-s + (1.18 + 0.993i)13-s + (2.15 + 1.81i)14-s + (2.67 + 0.974i)16-s + (2.66 − 4.61i)17-s + (−0.629 + 3.66i)18-s + (−2.28 − 3.96i)19-s + ⋯
L(s)  = 1  + (−0.670 + 0.562i)2-s + (0.938 − 0.344i)3-s + (−0.0405 + 0.229i)4-s + (−0.436 + 0.759i)6-s + (−0.149 − 0.847i)7-s + (−0.539 − 0.935i)8-s + (0.763 − 0.646i)9-s + (−1.46 − 0.534i)11-s + (0.0410 + 0.229i)12-s + (0.328 + 0.275i)13-s + (0.577 + 0.484i)14-s + (0.669 + 0.243i)16-s + (0.646 − 1.11i)17-s + (−0.148 + 0.862i)18-s + (−0.524 − 0.909i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.554 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.554 + 0.831i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.554 + 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.950277 - 0.508394i\)
\(L(\frac12)\) \(\approx\) \(0.950277 - 0.508394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 + 0.595i)T \)
5 \( 1 \)
good2 \( 1 + (0.948 - 0.795i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (0.395 + 2.24i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (4.87 + 1.77i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-1.18 - 0.993i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-2.66 + 4.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.28 + 3.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.892 + 5.06i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (4.94 - 4.15i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (0.228 - 1.29i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.84 - 3.18i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.26 - 2.74i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (-7.92 - 2.88i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (1.17 + 6.68i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 6.64T + 53T^{2} \)
59 \( 1 + (2.83 - 1.03i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.999 - 5.67i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.65 - 2.22i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (0.0130 - 0.0226i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.89 + 5.00i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-12.6 + 10.6i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-8.29 + 6.96i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (2.32 + 4.02i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.81 - 2.48i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.13818660092770017133334554517, −9.185330853416526864209489612950, −8.581591233316000295062631590681, −7.66269051268840405920203264328, −7.26878607416950659114431314410, −6.33227524453652314868405047686, −4.76997690198566663375179038140, −3.52030475104708951921049956932, −2.67333695211922877594818946765, −0.62956588922431442523141946175, 1.80165010618974213912188546853, 2.60206540706405814518910974875, 3.80901642371181204633044411700, 5.27808495982859634274631858944, 5.91829674706527412748158528787, 7.76238396291080019706690616821, 8.096675124452612885408999169446, 9.143022375366834381833065287986, 9.699022749164191741983015331227, 10.49431779327085479529879210075

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