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Properties

Label 2-300-100.23-c1-0-11
Degree $2$
Conductor $300$
Sign $0.302 - 0.953i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
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  + (−0.579 + 1.29i)2-s + (0.453 + 0.891i)3-s + (−1.32 − 1.49i)4-s + (1.07 − 1.96i)5-s + (−1.41 + 0.0697i)6-s + (0.868 + 0.868i)7-s + (2.69 − 0.849i)8-s + (−0.587 + 0.809i)9-s + (1.90 + 2.52i)10-s + (2.63 + 3.62i)11-s + (0.728 − 1.86i)12-s + (−0.530 − 3.34i)13-s + (−1.62 + 0.617i)14-s + (2.23 + 0.0672i)15-s + (−0.466 + 3.97i)16-s + (5.52 + 2.81i)17-s + ⋯
L(s)  = 1  + (−0.409 + 0.912i)2-s + (0.262 + 0.514i)3-s + (−0.664 − 0.747i)4-s + (0.480 − 0.876i)5-s + (−0.576 + 0.0284i)6-s + (0.328 + 0.328i)7-s + (0.953 − 0.300i)8-s + (−0.195 + 0.269i)9-s + (0.603 + 0.797i)10-s + (0.794 + 1.09i)11-s + (0.210 − 0.537i)12-s + (−0.147 − 0.928i)13-s + (−0.434 + 0.165i)14-s + (0.577 + 0.0173i)15-s + (−0.116 + 0.993i)16-s + (1.33 + 0.682i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.302 - 0.953i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.302 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01280 + 0.741307i\)
\(L(\frac12)\) \(\approx\) \(1.01280 + 0.741307i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.579 - 1.29i)T \)
3 \( 1 + (-0.453 - 0.891i)T \)
5 \( 1 + (-1.07 + 1.96i)T \)
good7 \( 1 + (-0.868 - 0.868i)T + 7iT^{2} \)
11 \( 1 + (-2.63 - 3.62i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.530 + 3.34i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-5.52 - 2.81i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (-1.47 - 4.52i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-0.748 + 4.72i)T + (-21.8 - 7.10i)T^{2} \)
29 \( 1 + (-2.21 - 0.720i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (9.32 - 3.02i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (8.45 - 1.33i)T + (35.1 - 11.4i)T^{2} \)
41 \( 1 + (-2.25 - 1.63i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-5.74 + 5.74i)T - 43iT^{2} \)
47 \( 1 + (-4.24 + 2.16i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-0.643 + 0.327i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (0.305 + 0.222i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.68 + 1.95i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.01 - 5.92i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (4.75 + 1.54i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.39 - 0.537i)T + (69.4 + 22.5i)T^{2} \)
79 \( 1 + (0.521 - 1.60i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (13.5 + 6.92i)T + (48.7 + 67.1i)T^{2} \)
89 \( 1 + (0.948 + 1.30i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (5.55 + 10.9i)T + (-57.0 + 78.4i)T^{2} \)
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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

−12.22564369385008666317390784158, −10.41459505528824835763900935247, −9.943539281955286043920129509721, −8.945842314943590230202535158670, −8.288909873342061458062073117988, −7.24828520702109554052636742413, −5.77129063521090441189426680636, −5.15902697389530587167673147356, −3.95187008202201197026313405176, −1.58669365255660721309385089872, 1.35297315927341223274497289470, 2.80795706966470580615630995881, 3.80932251526048568931337324596, 5.55011746208427393421048517343, 6.99324025917163579522224956473, 7.67797671051661416794637174456, 9.072282024476679444625262809823, 9.537589047814516260835165822855, 10.87430800791283714781099969759, 11.38602306002361423592116738289

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