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Properties

Label 2-3675-15.14-c0-0-11
Degree $2$
Conductor $3675$
Sign $-0.316 + 0.948i$
Analytic cond. $1.83406$
Root an. cond. $1.35427$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + (0.707 − 0.707i)3-s + (0.707 − 0.707i)6-s − 8-s − 1.00i·9-s − i·11-s − 1.41i·13-s − 16-s − 1.41·17-s − 1.00i·18-s − i·22-s + 23-s + (−0.707 + 0.707i)24-s − 1.41i·26-s + (−0.707 − 0.707i)27-s + ⋯
L(s)  = 1  + 2-s + (0.707 − 0.707i)3-s + (0.707 − 0.707i)6-s − 8-s − 1.00i·9-s − i·11-s − 1.41i·13-s − 16-s − 1.41·17-s − 1.00i·18-s − i·22-s + 23-s + (−0.707 + 0.707i)24-s − 1.41i·26-s + (−0.707 − 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.316 + 0.948i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (2549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ -0.316 + 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.019232012\)
\(L(\frac12)\) \(\approx\) \(2.019232012\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 - T + T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 - iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT - T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41iT - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 + iT - T^{2} \)
71 \( 1 + iT - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.561778164716338305902066072440, −7.79575681904382771832224284145, −6.79373269364118795539301154120, −6.27399025797627996991070984045, −5.41198716296725144750068194413, −4.73028675993986301333392718250, −3.54241573000367919136930449852, −3.19629348921094898149483370976, −2.28833442704790488035368851254, −0.74430272196132175298441590229, 2.03289039338714900240062601783, 2.67745929481078779548918971847, 3.83109395389377647197545016913, 4.36904262518443610069134769195, 4.76403826014019186605322703975, 5.70673311388995692299048142228, 6.71996137400106491116957929461, 7.26029199277723352385276479401, 8.445483156448179180100285991983, 9.004010692094912627080227857349

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