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Properties

Label 2-170-85.12-c1-0-7
Degree $2$
Conductor $170$
Sign $-0.446 + 0.894i$
Analytic cond. $1.35745$
Root an. cond. $1.16509$
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  + (0.382 − 0.923i)2-s + (0.0178 − 0.0897i)3-s + (−0.707 − 0.707i)4-s + (−0.936 − 2.03i)5-s + (−0.0760 − 0.0508i)6-s + (−3.86 − 2.58i)7-s + (−0.923 + 0.382i)8-s + (2.76 + 1.14i)9-s + (−2.23 + 0.0883i)10-s + (1.32 − 1.98i)11-s + (−0.0760 + 0.0508i)12-s + 4.39·13-s + (−3.86 + 2.58i)14-s + (−0.198 + 0.0478i)15-s + i·16-s + (−2.01 + 3.59i)17-s + ⋯
L(s)  = 1  + (0.270 − 0.653i)2-s + (0.0103 − 0.0518i)3-s + (−0.353 − 0.353i)4-s + (−0.418 − 0.908i)5-s + (−0.0310 − 0.0207i)6-s + (−1.46 − 0.976i)7-s + (−0.326 + 0.135i)8-s + (0.921 + 0.381i)9-s + (−0.706 + 0.0279i)10-s + (0.399 − 0.598i)11-s + (−0.0219 + 0.0146i)12-s + 1.21·13-s + (−1.03 + 0.690i)14-s + (−0.0513 + 0.0123i)15-s + 0.250i·16-s + (−0.489 + 0.872i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.446 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.446 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(170\)    =    \(2 \cdot 5 \cdot 17\)
Sign: $-0.446 + 0.894i$
Analytic conductor: \(1.35745\)
Root analytic conductor: \(1.16509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{170} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 170,\ (\ :1/2),\ -0.446 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.578132 - 0.935050i\)
\(L(\frac12)\) \(\approx\) \(0.578132 - 0.935050i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (0.936 + 2.03i)T \)
17 \( 1 + (2.01 - 3.59i)T \)
good3 \( 1 + (-0.0178 + 0.0897i)T + (-2.77 - 1.14i)T^{2} \)
7 \( 1 + (3.86 + 2.58i)T + (2.67 + 6.46i)T^{2} \)
11 \( 1 + (-1.32 + 1.98i)T + (-4.20 - 10.1i)T^{2} \)
13 \( 1 - 4.39T + 13T^{2} \)
19 \( 1 + (-3.95 + 1.63i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (0.00109 - 0.000217i)T + (21.2 - 8.80i)T^{2} \)
29 \( 1 + (3.34 + 0.664i)T + (26.7 + 11.0i)T^{2} \)
31 \( 1 + (-2.20 - 3.30i)T + (-11.8 + 28.6i)T^{2} \)
37 \( 1 + (-3.12 - 0.622i)T + (34.1 + 14.1i)T^{2} \)
41 \( 1 + (-8.42 + 1.67i)T + (37.8 - 15.6i)T^{2} \)
43 \( 1 + (-0.627 - 1.51i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 12.7iT - 47T^{2} \)
53 \( 1 + (3.98 + 1.64i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-0.521 + 1.25i)T + (-41.7 - 41.7i)T^{2} \)
61 \( 1 + (-2.18 - 10.9i)T + (-56.3 + 23.3i)T^{2} \)
67 \( 1 + (4.34 - 4.34i)T - 67iT^{2} \)
71 \( 1 + (6.21 - 4.15i)T + (27.1 - 65.5i)T^{2} \)
73 \( 1 + (-8.62 + 5.76i)T + (27.9 - 67.4i)T^{2} \)
79 \( 1 + (7.53 + 5.03i)T + (30.2 + 72.9i)T^{2} \)
83 \( 1 + (5.65 - 13.6i)T + (-58.6 - 58.6i)T^{2} \)
89 \( 1 + (10.3 + 10.3i)T + 89iT^{2} \)
97 \( 1 + (2.50 - 1.67i)T + (37.1 - 89.6i)T^{2} \)
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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

−12.75165810993989012286261716805, −11.47803576096272058738753684733, −10.50565846153629787550340607910, −9.569339520622521433908672281484, −8.573112796763967114697931754088, −7.15659618120527240401921288723, −5.93247608723996046278833702608, −4.26179954871559237044524047380, −3.54069654089804831901259134587, −1.07035889741505614401869283069, 3.01551867018314410069571260090, 4.11227036557054943487921336880, 5.98119734900842687122513599160, 6.65807966073192593058092820353, 7.62549925943838617058792082308, 9.223310910127385855037903932958, 9.751562645118836038758089235230, 11.25521726008798338153611555161, 12.31558420460327369367069664155, 13.06629417841643101509647391797

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