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Properties

Label 2-675-45.14-c2-0-31
Degree $2$
Conductor $675$
Sign $-0.824 + 0.565i$
Analytic cond. $18.3924$
Root an. cond. $4.28863$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.696 + 1.20i)2-s + (1.03 − 1.78i)4-s + (−7.63 + 4.41i)7-s + 8.43·8-s + (0.805 − 0.464i)11-s + (−21.2 − 12.2i)13-s + (−10.6 − 6.13i)14-s + (1.74 + 3.02i)16-s − 18.3·17-s + 5.58·19-s + (1.12 + 0.647i)22-s + (−11.9 + 20.6i)23-s − 34.0i·26-s + 18.1i·28-s + (−23.7 + 13.7i)29-s + ⋯
L(s)  = 1  + (0.348 + 0.602i)2-s + (0.257 − 0.446i)4-s + (−1.09 + 0.630i)7-s + 1.05·8-s + (0.0732 − 0.0422i)11-s + (−1.63 − 0.941i)13-s + (−0.759 − 0.438i)14-s + (0.109 + 0.189i)16-s − 1.07·17-s + 0.294·19-s + (0.0509 + 0.0294i)22-s + (−0.517 + 0.896i)23-s − 1.31i·26-s + 0.649i·28-s + (−0.819 + 0.473i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.824 + 0.565i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.824 + 0.565i$
Analytic conductor: \(18.3924\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1),\ -0.824 + 0.565i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1506321194\)
\(L(\frac12)\) \(\approx\) \(0.1506321194\)
\(L(2)\) 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 \( 1 + (-0.696 - 1.20i)T + (-2 + 3.46i)T^{2} \)
7 \( 1 + (7.63 - 4.41i)T + (24.5 - 42.4i)T^{2} \)
11 \( 1 + (-0.805 + 0.464i)T + (60.5 - 104. i)T^{2} \)
13 \( 1 + (21.2 + 12.2i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 + 18.3T + 289T^{2} \)
19 \( 1 - 5.58T + 361T^{2} \)
23 \( 1 + (11.9 - 20.6i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (23.7 - 13.7i)T + (420.5 - 728. i)T^{2} \)
31 \( 1 + (-4.66 + 8.08i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + 24.7iT - 1.36e3T^{2} \)
41 \( 1 + (-6.45 - 3.72i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (30.7 - 17.7i)T + (924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-0.172 - 0.298i)T + (-1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 81.8T + 2.80e3T^{2} \)
59 \( 1 + (65.9 + 38.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (29.6 + 51.3i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-70.9 - 40.9i)T + (2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 37.5iT - 5.04e3T^{2} \)
73 \( 1 + 3.49iT - 5.32e3T^{2} \)
79 \( 1 + (62.0 + 107. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-27.9 - 48.4i)T + (-3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 6.78iT - 7.92e3T^{2} \)
97 \( 1 + (-101. + 58.5i)T + (4.70e3 - 8.14e3i)T^{2} \)
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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

−9.766110147157446759258681358235, −9.351984841544267687092932866092, −7.917358146146191656869907994642, −7.15990391192577323320957483094, −6.28957504387482313140867798561, −5.53348761507860963527001844167, −4.68946882701209429706834009920, −3.20071984729527856730092978768, −2.06055297309524508479067609256, −0.04245747338016859779697703671, 2.00597193935420619084203380854, 2.96740349663810338688156615019, 4.07198093147985088530659129609, 4.77793714321100216255546507526, 6.46937230783043538979741465846, 7.01863341350541665717996982078, 7.88046269260497990007981671534, 9.167158716557622715437834685712, 9.915332931279917746314594697650, 10.67326266093377075062562830318

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