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Properties

Label 2-176-16.13-c1-0-4
Degree $2$
Conductor $176$
Sign $-0.490 - 0.871i$
Analytic cond. $1.40536$
Root an. cond. $1.18548$
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  + (−1.15 + 0.820i)2-s + (2.16 + 2.16i)3-s + (0.652 − 1.89i)4-s + (−1.53 + 1.53i)5-s + (−4.26 − 0.715i)6-s − 0.310i·7-s + (0.799 + 2.71i)8-s + 6.33i·9-s + (0.508 − 3.02i)10-s + (0.707 − 0.707i)11-s + (5.49 − 2.67i)12-s + (−1.66 − 1.66i)13-s + (0.254 + 0.357i)14-s − 6.63·15-s + (−3.14 − 2.46i)16-s + 7.00·17-s + ⋯
L(s)  = 1  + (−0.814 + 0.580i)2-s + (1.24 + 1.24i)3-s + (0.326 − 0.945i)4-s + (−0.686 + 0.686i)5-s + (−1.73 − 0.292i)6-s − 0.117i·7-s + (0.282 + 0.959i)8-s + 2.11i·9-s + (0.160 − 0.957i)10-s + (0.213 − 0.213i)11-s + (1.58 − 0.771i)12-s + (−0.462 − 0.462i)13-s + (0.0680 + 0.0954i)14-s − 1.71·15-s + (−0.786 − 0.617i)16-s + 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176\)    =    \(2^{4} \cdot 11\)
Sign: $-0.490 - 0.871i$
Analytic conductor: \(1.40536\)
Root analytic conductor: \(1.18548\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{176} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 176,\ (\ :1/2),\ -0.490 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528335 + 0.903925i\)
\(L(\frac12)\) \(\approx\) \(0.528335 + 0.903925i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.15 - 0.820i)T \)
11 \( 1 + (-0.707 + 0.707i)T \)
good3 \( 1 + (-2.16 - 2.16i)T + 3iT^{2} \)
5 \( 1 + (1.53 - 1.53i)T - 5iT^{2} \)
7 \( 1 + 0.310iT - 7T^{2} \)
13 \( 1 + (1.66 + 1.66i)T + 13iT^{2} \)
17 \( 1 - 7.00T + 17T^{2} \)
19 \( 1 + (3.63 + 3.63i)T + 19iT^{2} \)
23 \( 1 - 5.24iT - 23T^{2} \)
29 \( 1 + (4.00 + 4.00i)T + 29iT^{2} \)
31 \( 1 - 5.00T + 31T^{2} \)
37 \( 1 + (-6.72 + 6.72i)T - 37iT^{2} \)
41 \( 1 + 1.04iT - 41T^{2} \)
43 \( 1 + (-2.56 + 2.56i)T - 43iT^{2} \)
47 \( 1 + 4.16T + 47T^{2} \)
53 \( 1 + (-6.07 + 6.07i)T - 53iT^{2} \)
59 \( 1 + (5.67 - 5.67i)T - 59iT^{2} \)
61 \( 1 + (-7.98 - 7.98i)T + 61iT^{2} \)
67 \( 1 + (8.03 + 8.03i)T + 67iT^{2} \)
71 \( 1 - 1.26iT - 71T^{2} \)
73 \( 1 - 1.03iT - 73T^{2} \)
79 \( 1 + 9.33T + 79T^{2} \)
83 \( 1 + (1.59 + 1.59i)T + 83iT^{2} \)
89 \( 1 + 7.57iT - 89T^{2} \)
97 \( 1 + 1.71T + 97T^{2} \)
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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

−13.50483750361760254583142877172, −11.58255986301841250618799847001, −10.60036314721903468552025882045, −9.873786644801593928327806405002, −9.036933540913242593327419904642, −7.952458196709187631418352460246, −7.36179330336670637912222518025, −5.53648203546081211353184528587, −4.02671515441585043014843482623, −2.76776834338927969728748019535, 1.27550155177832416704799162390, 2.71756891687393135109433317881, 4.06991298238074153986855987553, 6.58653901478441119868472677959, 7.74727250260573029094853946109, 8.199041686141710371141408913174, 9.075654037861684784978407008076, 10.11304250180996416042619566740, 11.81800469715744231315581335023, 12.36672557670613906555194964075

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