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Properties

Label 2-675-45.23-c1-0-11
Degree $2$
Conductor $675$
Sign $0.736 + 0.676i$
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  + (−0.601 + 2.24i)2-s + (−2.93 − 1.69i)4-s + (−0.751 − 0.201i)7-s + (2.29 − 2.29i)8-s + (0.220 − 0.127i)11-s + (−3.70 + 0.992i)13-s + (0.903 − 1.56i)14-s + (0.367 + 0.636i)16-s + (−3.93 − 3.93i)17-s − 0.440i·19-s + (0.152 + 0.570i)22-s + (−0.917 − 3.42i)23-s − 8.90i·26-s + (1.86 + 1.86i)28-s + (−2.76 − 4.78i)29-s + ⋯
L(s)  = 1  + (−0.425 + 1.58i)2-s + (−1.46 − 0.848i)4-s + (−0.284 − 0.0761i)7-s + (0.809 − 0.809i)8-s + (0.0663 − 0.0383i)11-s + (−1.02 + 0.275i)13-s + (0.241 − 0.418i)14-s + (0.0918 + 0.159i)16-s + (−0.953 − 0.953i)17-s − 0.101i·19-s + (0.0325 + 0.121i)22-s + (−0.191 − 0.713i)23-s − 1.74i·26-s + (0.352 + 0.352i)28-s + (−0.513 − 0.888i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.736 + 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.736 + 0.676i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (368, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.736 + 0.676i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.250263 - 0.0974265i\)
\(L(\frac12)\) \(\approx\) \(0.250263 - 0.0974265i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.601 - 2.24i)T + (-1.73 - i)T^{2} \)
7 \( 1 + (0.751 + 0.201i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (-0.220 + 0.127i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.70 - 0.992i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (3.93 + 3.93i)T + 17iT^{2} \)
19 \( 1 + 0.440iT - 19T^{2} \)
23 \( 1 + (0.917 + 3.42i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (2.76 + 4.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.0971 - 0.168i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.123 + 0.123i)T - 37iT^{2} \)
41 \( 1 + (-3.88 - 2.24i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.357 + 1.33i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (-1.11 + 4.17i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.938 - 0.938i)T - 53iT^{2} \)
59 \( 1 + (4.02 - 6.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.44 + 2.49i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.47 + 12.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.15iT - 71T^{2} \)
73 \( 1 + (-9.18 - 9.18i)T + 73iT^{2} \)
79 \( 1 + (11.9 - 6.91i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.20 + 1.39i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 - 0.285T + 89T^{2} \)
97 \( 1 + (8.73 + 2.34i)T + (84.0 + 48.5i)T^{2} \)
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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

−9.910845733080660325082585175977, −9.356239569833233058251827832025, −8.535705288563681107745272702195, −7.59241438830123146012507914997, −6.93413548157910681881771943300, −6.19642253119270092407764800535, −5.13592502753823118816228050857, −4.31985725506761277617147227037, −2.53650647156151409422082593484, −0.15951261929397770079789293689, 1.62576467622380471474023168281, 2.69990038737128596726905039794, 3.72649246609654686935474578584, 4.73217523648507830395114638609, 6.06357844851004160304698801878, 7.29527461016412631571568106604, 8.362170000526084480988908777535, 9.239595781731950543360780135009, 9.825686654198404358924202773220, 10.69038024311076181640874435695

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