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Properties

Label 2-30e2-25.11-c1-0-1
Degree $2$
Conductor $900$
Sign $-0.604 - 0.796i$
Analytic cond. $7.18653$
Root an. cond. $2.68077$
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  + (1.49 + 1.66i)5-s − 4.78·7-s + (1.58 − 1.14i)11-s + (−0.873 − 0.634i)13-s + (1.17 + 3.61i)17-s + (1.31 + 4.04i)19-s + (−4.74 + 3.44i)23-s + (−0.522 + 4.97i)25-s + (−3.26 + 10.0i)29-s + (−1.33 − 4.10i)31-s + (−7.15 − 7.94i)35-s + (−4.57 − 3.32i)37-s + (0.694 + 0.504i)41-s − 10.8·43-s + (−0.927 + 2.85i)47-s + ⋯
L(s)  = 1  + (0.669 + 0.743i)5-s − 1.80·7-s + (0.477 − 0.346i)11-s + (−0.242 − 0.176i)13-s + (0.285 + 0.877i)17-s + (0.301 + 0.927i)19-s + (−0.989 + 0.718i)23-s + (−0.104 + 0.994i)25-s + (−0.605 + 1.86i)29-s + (−0.239 − 0.737i)31-s + (−1.20 − 1.34i)35-s + (−0.752 − 0.546i)37-s + (0.108 + 0.0788i)41-s − 1.65·43-s + (−0.135 + 0.416i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.604 - 0.796i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ -0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.396304 + 0.798350i\)
\(L(\frac12)\) \(\approx\) \(0.396304 + 0.798350i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-1.49 - 1.66i)T \)
good7 \( 1 + 4.78T + 7T^{2} \)
11 \( 1 + (-1.58 + 1.14i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (0.873 + 0.634i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.17 - 3.61i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.31 - 4.04i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (4.74 - 3.44i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (3.26 - 10.0i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.33 + 4.10i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (4.57 + 3.32i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-0.694 - 0.504i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 10.8T + 43T^{2} \)
47 \( 1 + (0.927 - 2.85i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (1.30 - 4.01i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.85 + 2.80i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.93 - 2.13i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-2.14 - 6.59i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-3.70 + 11.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-13.7 + 9.96i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.04 - 6.29i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.797 - 2.45i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.673 - 0.489i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (2.81 - 8.67i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.22159675916632831236769690335, −9.698849157988490859808234319360, −9.007178847420638264639693267629, −7.73757419271364039999085383440, −6.80346661483059680222614992052, −6.14193386449421524950987012265, −5.51957863748972840728183648195, −3.61993620982835888827964359347, −3.29990854050427027932397915327, −1.80874729794306937267593840307, 0.40002406496900186067314329014, 2.17614025498434116938283831837, 3.30985239778324370172691487678, 4.46783854682799277461747572402, 5.49813121414092445850620412476, 6.44537456220484486556181568471, 6.97898977977135846450866102966, 8.290917978135032383022010576443, 9.301578867211627064057062886604, 9.673010170252311806951974844236

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