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Properties

Label 2-750-125.91-c1-0-21
Degree $2$
Conductor $750$
Sign $0.261 + 0.965i$
Analytic cond. $5.98878$
Root an. cond. $2.44719$
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  + (0.728 + 0.684i)2-s + (0.535 − 0.844i)3-s + (0.0627 + 0.998i)4-s + (−0.583 − 2.15i)5-s + (0.968 − 0.248i)6-s + (−0.444 − 1.36i)7-s + (−0.637 + 0.770i)8-s + (−0.425 − 0.904i)9-s + (1.05 − 1.97i)10-s + (−0.0647 − 0.0608i)11-s + (0.876 + 0.481i)12-s + (−0.661 − 1.40i)13-s + (0.612 − 1.30i)14-s + (−2.13 − 0.663i)15-s + (−0.992 + 0.125i)16-s + (0.380 − 6.04i)17-s + ⋯
L(s)  = 1  + (0.515 + 0.484i)2-s + (0.309 − 0.487i)3-s + (0.0313 + 0.499i)4-s + (−0.261 − 0.965i)5-s + (0.395 − 0.101i)6-s + (−0.167 − 0.516i)7-s + (−0.225 + 0.272i)8-s + (−0.141 − 0.301i)9-s + (0.332 − 0.623i)10-s + (−0.0195 − 0.0183i)11-s + (0.252 + 0.139i)12-s + (−0.183 − 0.389i)13-s + (0.163 − 0.347i)14-s + (−0.551 − 0.171i)15-s + (−0.248 + 0.0313i)16-s + (0.0921 − 1.46i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 750 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(750\)    =    \(2 \cdot 3 \cdot 5^{3}\)
Sign: $0.261 + 0.965i$
Analytic conductor: \(5.98878\)
Root analytic conductor: \(2.44719\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{750} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 750,\ (\ :1/2),\ 0.261 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.43504 - 1.09776i\)
\(L(\frac12)\) \(\approx\) \(1.43504 - 1.09776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.728 - 0.684i)T \)
3 \( 1 + (-0.535 + 0.844i)T \)
5 \( 1 + (0.583 + 2.15i)T \)
good7 \( 1 + (0.444 + 1.36i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (0.0647 + 0.0608i)T + (0.690 + 10.9i)T^{2} \)
13 \( 1 + (0.661 + 1.40i)T + (-8.28 + 10.0i)T^{2} \)
17 \( 1 + (-0.380 + 6.04i)T + (-16.8 - 2.13i)T^{2} \)
19 \( 1 + (2.22 + 3.51i)T + (-8.08 + 17.1i)T^{2} \)
23 \( 1 + (-1.46 - 7.68i)T + (-21.3 + 8.46i)T^{2} \)
29 \( 1 + (-6.11 + 2.41i)T + (21.1 - 19.8i)T^{2} \)
31 \( 1 + (-0.584 + 9.28i)T + (-30.7 - 3.88i)T^{2} \)
37 \( 1 + (-0.386 + 0.0488i)T + (35.8 - 9.20i)T^{2} \)
41 \( 1 + (0.426 - 2.23i)T + (-38.1 - 15.0i)T^{2} \)
43 \( 1 + (2.33 - 1.69i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (2.13 + 2.58i)T + (-8.80 + 46.1i)T^{2} \)
53 \( 1 + (3.61 + 0.927i)T + (46.4 + 25.5i)T^{2} \)
59 \( 1 + (-11.0 - 6.06i)T + (31.6 + 49.8i)T^{2} \)
61 \( 1 + (-0.957 - 5.02i)T + (-56.7 + 22.4i)T^{2} \)
67 \( 1 + (-5.86 - 2.32i)T + (48.8 + 45.8i)T^{2} \)
71 \( 1 + (-6.34 - 7.66i)T + (-13.3 + 69.7i)T^{2} \)
73 \( 1 + (3.74 - 2.05i)T + (39.1 - 61.6i)T^{2} \)
79 \( 1 + (-2.70 + 4.26i)T + (-33.6 - 71.4i)T^{2} \)
83 \( 1 + (-2.15 - 3.38i)T + (-35.3 + 75.1i)T^{2} \)
89 \( 1 + (6.39 - 3.51i)T + (47.6 - 75.1i)T^{2} \)
97 \( 1 + (-13.5 + 5.34i)T + (70.7 - 66.4i)T^{2} \)
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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

−9.925626386885705042631144261559, −9.193028766132909469193163655185, −8.247444507741727637873925948116, −7.53171656524099248667658583136, −6.82317312146376805792599540785, −5.61913415842302701876146831235, −4.79368722576390222009457730457, −3.80293423132155592335821618076, −2.56942040080139560661000148951, −0.75292314360689627147172421870, 2.03490082449380165268093004261, 3.06243254026206481050686697015, 3.91394559090488084573004251116, 4.90937892863679118829340155770, 6.17192806399507764198926514213, 6.75530704963140981239941319680, 8.171279832964337670785999126244, 8.815942589516844543597972345120, 10.10049228778011394285619255136, 10.46469128277416652229426259061

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