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Properties

Label 2-3675-3.2-c0-0-11
Degree $2$
Conductor $3675$
Sign $0.707 + 0.707i$
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  + i·2-s + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)6-s + i·8-s + 1.00i·9-s − i·11-s − 1.41·13-s − 16-s − 1.41i·17-s − 1.00·18-s + 22-s − i·23-s + (0.707 − 0.707i)24-s − 1.41i·26-s + (0.707 − 0.707i)27-s + ⋯
L(s)  = 1  + i·2-s + (−0.707 − 0.707i)3-s + (0.707 − 0.707i)6-s + i·8-s + 1.00i·9-s − i·11-s − 1.41·13-s − 16-s − 1.41i·17-s − 1.00·18-s + 22-s − i·23-s + (0.707 − 0.707i)24-s − 1.41i·26-s + (0.707 − 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3675\)    =    \(3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(1.83406\)
Root analytic conductor: \(1.35427\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3675} (1226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3675,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7896943591\)
\(L(\frac12)\) \(\approx\) \(0.7896943591\)
\(L(1)\) 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 \( 1 - iT - T^{2} \)
11 \( 1 + iT - T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 + 1.41iT - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT - T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 1.41iT - T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 - T + T^{2} \)
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

−8.352005890595209765229862333634, −7.55429249207595863013666283175, −7.13955352984871371197444190915, −6.52251198184869712923441683110, −5.67129310899660305364519882397, −5.24841270692161058275100503825, −4.41285571622442849484471061490, −2.79705773849163544078853760342, −2.18548042375039198488974853812, −0.48696297733850046825537053105, 1.38141272191104180840839095124, 2.32429909025351168891522701803, 3.38776931728900539436148658246, 4.08519432226269481088332792891, 4.87441722709380133756435154013, 5.61334381343422161538168226671, 6.68023529633621945488448444678, 7.10525722706810537438235774703, 8.126765232203114322265202665526, 9.301573430995994763077160572316

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