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Properties

Label 2-350-175.17-c1-0-14
Degree $2$
Conductor $350$
Sign $0.835 + 0.550i$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
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  + (−0.998 + 0.0523i)2-s + (1.80 − 2.22i)3-s + (0.994 − 0.104i)4-s + (0.724 + 2.11i)5-s + (−1.68 + 2.32i)6-s + (2.64 + 0.0482i)7-s + (−0.987 + 0.156i)8-s + (−1.08 − 5.11i)9-s + (−0.834 − 2.07i)10-s + (4.46 + 0.949i)11-s + (1.56 − 2.40i)12-s + (−3.02 + 1.54i)13-s + (−2.64 + 0.0902i)14-s + (6.02 + 2.20i)15-s + (0.978 − 0.207i)16-s + (−2.61 + 6.80i)17-s + ⋯
L(s)  = 1  + (−0.706 + 0.0370i)2-s + (1.04 − 1.28i)3-s + (0.497 − 0.0522i)4-s + (0.323 + 0.946i)5-s + (−0.688 + 0.947i)6-s + (0.999 + 0.0182i)7-s + (−0.349 + 0.0553i)8-s + (−0.362 − 1.70i)9-s + (−0.263 − 0.656i)10-s + (1.34 + 0.286i)11-s + (0.450 − 0.694i)12-s + (−0.839 + 0.427i)13-s + (−0.706 + 0.0241i)14-s + (1.55 + 0.568i)15-s + (0.244 − 0.0519i)16-s + (−0.633 + 1.64i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.835 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $0.835 + 0.550i$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{350} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 0.835 + 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49058 - 0.446863i\)
\(L(\frac12)\) \(\approx\) \(1.49058 - 0.446863i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.998 - 0.0523i)T \)
5 \( 1 + (-0.724 - 2.11i)T \)
7 \( 1 + (-2.64 - 0.0482i)T \)
good3 \( 1 + (-1.80 + 2.22i)T + (-0.623 - 2.93i)T^{2} \)
11 \( 1 + (-4.46 - 0.949i)T + (10.0 + 4.47i)T^{2} \)
13 \( 1 + (3.02 - 1.54i)T + (7.64 - 10.5i)T^{2} \)
17 \( 1 + (2.61 - 6.80i)T + (-12.6 - 11.3i)T^{2} \)
19 \( 1 + (-0.199 + 1.90i)T + (-18.5 - 3.95i)T^{2} \)
23 \( 1 + (0.292 + 5.58i)T + (-22.8 + 2.40i)T^{2} \)
29 \( 1 + (0.582 + 0.801i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (3.63 + 8.16i)T + (-20.7 + 23.0i)T^{2} \)
37 \( 1 + (8.71 + 5.66i)T + (15.0 + 33.8i)T^{2} \)
41 \( 1 + (-0.915 - 0.297i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + (-1.12 + 1.12i)T - 43iT^{2} \)
47 \( 1 + (2.83 - 1.08i)T + (34.9 - 31.4i)T^{2} \)
53 \( 1 + (2.02 + 1.63i)T + (11.0 + 51.8i)T^{2} \)
59 \( 1 + (-7.12 - 7.91i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (2.45 + 2.21i)T + (6.37 + 60.6i)T^{2} \)
67 \( 1 + (9.43 + 3.62i)T + (49.7 + 44.8i)T^{2} \)
71 \( 1 + (8.04 - 5.84i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.79 - 7.38i)T + (-29.6 + 66.6i)T^{2} \)
79 \( 1 + (1.05 - 2.37i)T + (-52.8 - 58.7i)T^{2} \)
83 \( 1 + (1.33 + 8.45i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (0.674 - 0.748i)T + (-9.30 - 88.5i)T^{2} \)
97 \( 1 + (0.741 - 4.68i)T + (-92.2 - 29.9i)T^{2} \)
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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

−11.41943704911704369265004521780, −10.43765262582722713552873601346, −9.215578930586327234222043869042, −8.576291128818313741394188286520, −7.57690291486580197201668546026, −6.94007843958881858767041193686, −6.15474912026924843909793945143, −3.98894665677356725831723750465, −2.32912845900100912300511585387, −1.73265869238621453181025525095, 1.69983086488623658773176688762, 3.26472831510753440783104717022, 4.56709321680805958480838575526, 5.31855253140451390934483551187, 7.19815231601374129880081850138, 8.278643155069259374894065060126, 8.999126037797064157632379672796, 9.439065539568923616341818636695, 10.30124830942715594438784036194, 11.41876794189214417185383995263

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