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Properties

Label 2-30e2-20.7-c1-0-35
Degree $2$
Conductor $900$
Sign $0.787 + 0.616i$
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  + (1.40 − 0.178i)2-s + (1.93 − 0.5i)4-s + (2.62 − 1.04i)8-s − 3.87i·11-s + (2.44 − 2.44i)13-s + (3.50 − 1.93i)16-s + (1.22 + 1.22i)17-s − 3.87·19-s + (−0.690 − 5.43i)22-s + (3.16 + 3.16i)23-s + (3 − 3.87i)26-s + 6i·29-s − 7.74i·31-s + (4.56 − 3.34i)32-s + (1.93 + 1.5i)34-s + ⋯
L(s)  = 1  + (0.992 − 0.126i)2-s + (0.968 − 0.250i)4-s + (0.929 − 0.370i)8-s − 1.16i·11-s + (0.679 − 0.679i)13-s + (0.875 − 0.484i)16-s + (0.297 + 0.297i)17-s − 0.888·19-s + (−0.147 − 1.15i)22-s + (0.659 + 0.659i)23-s + (0.588 − 0.759i)26-s + 1.11i·29-s − 1.39i·31-s + (0.807 − 0.590i)32-s + (0.332 + 0.257i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(900\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(7.18653\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{900} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 900,\ (\ :1/2),\ 0.787 + 0.616i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.91085 - 1.00332i\)
\(L(\frac12)\) \(\approx\) \(2.91085 - 1.00332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.40 + 0.178i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 + 3.87iT - 11T^{2} \)
13 \( 1 + (-2.44 + 2.44i)T - 13iT^{2} \)
17 \( 1 + (-1.22 - 1.22i)T + 17iT^{2} \)
19 \( 1 + 3.87T + 19T^{2} \)
23 \( 1 + (-3.16 - 3.16i)T + 23iT^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 7.74iT - 31T^{2} \)
37 \( 1 + (-4.89 - 4.89i)T + 37iT^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + (-3.16 + 3.16i)T - 47iT^{2} \)
53 \( 1 + (2.44 - 2.44i)T - 53iT^{2} \)
59 \( 1 + 7.74T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + (4.74 - 4.74i)T - 67iT^{2} \)
71 \( 1 + 7.74iT - 71T^{2} \)
73 \( 1 + (3.67 - 3.67i)T - 73iT^{2} \)
79 \( 1 + 15.4T + 79T^{2} \)
83 \( 1 + (-1.58 - 1.58i)T + 83iT^{2} \)
89 \( 1 + 9iT - 89T^{2} \)
97 \( 1 + (-4.89 - 4.89i)T + 97iT^{2} \)
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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

−10.40413614797512562358048575250, −9.182296726401431885546014832131, −8.206793880074903673349394202064, −7.39119652981472331327926012283, −6.14565295307813836947105761641, −5.81715732461990793872599524153, −4.63585939445562329374952263236, −3.59568585002359984586605054059, −2.80879286502934210706224529082, −1.23642594833769656899458938978, 1.72437490514850322643394486309, 2.83728771952289580122682080659, 4.14086167318936533912933751578, 4.68882548775442173370587330812, 5.85653551810891572604426364850, 6.68497198407540712673914791810, 7.37719218191335740362364384229, 8.388830805898799261072882301780, 9.399010626185209609302405726195, 10.43493081741606624222418468058

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